Exercise 19.12
Question 1
(i) $\int \frac{d x}{\sqrt{1+4 x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{1+4 x^{2}}}=\int \frac{d x}{\sqrt{(1)^{2}+(2 x)^{2}}}$
$=\int \frac{d x}{\sqrt{1+(2 x)^{2}}}$
$I=\frac{1}{2} \log (2 x+\sqrt{1+4 x^{2}})$ $ \because \int \frac{d x}{\sqrt{a^{2}+x^{2}}}=\log (x+\sqrt{x^{2}+a^{2}})$
$I=\frac{1}{2} \log (2 x+\sqrt{1+4 x^{2}})+c$
(ii) $\int \frac{d x}{\sqrt{9-25 x^{2}}}$
Sol :
$I=\int \frac{d x}{\sqrt{9-25 x^{2}}}=\int \frac{d x}{\sqrt{(3)^{2}-(5 x)^{2}}}$
$\left[\because \int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1} \frac{x}{a}\right]$
$I=\frac{1}{5} \sin ^{-1} \frac{5 x}{3}+c$
Question 2
(i) $\int \frac{d x}{\sqrt{x^{2}+3 x+4}}$Sol :
$I=\int \frac{d x}{\sqrt{x^{2}+3 x+1}}$
$=\int \frac{d x}{\sqrt{x^{2}+2 x \cdot \frac{3}{2}+\frac{9}{4}-\frac{9}{4}}+4}$
$I=\int \frac{d x}{\sqrt{x^{2}+2 \cdot x \cdot \frac{3}{2}+\frac{9}{2}+7}}$
$=\int \frac{d x}{\sqrt{\left(x+\frac{3}{2}\right)^{2}+\left(\frac{\sqrt{7}}{2}\right)^{2}}}$
$I=\log \left[x+\frac{3}{2}+\sqrt{(x+2)^{2}+\left(\frac{\sqrt{7}}{2}\right)^{2}}\right]$
$I=\log \left[x+\frac{3}{2}+\sqrt{x^{2}+3 x+4}\right]+c$
(ii) $\int \frac{d x}{2 a x-x^{2}}$
Sol :
$I=\int \frac{d x}{\sqrt{2 a x-x^{2}}}$
$=\int \frac{d x}{\sqrt{2 a x \cdot x^{2}-a^{2}+a^{2}}}$
$I=\int \frac{d x}{\sqrt{a^{2}-\left(x^{2}+a^{2}-2 a x\right)}}$
$=\int \frac{d x}{\sqrt{a^{2}-(x-a)^{2}}}$
$I=\sin ^{-1} \frac{(x-a)}{a}+c$
Question 3
$\int \sqrt{3 x^{2}+4x+1} d x$Sol :
$I=\int \sqrt{3 x^{2}+4x+1} d x$
$=\int \sqrt{3 x^{2}+\frac{3}{3} 4 x+\frac{3}{3}} d x$
$I=\int \sqrt{3\left(x^{2}+\frac{4 x}{3}+\frac{1}{3}\right)}$
$=\sqrt{3} \int \sqrt{ x^{2}+\frac{4 x}{3}+\frac{1}{3}} d x$
$I=\sqrt{3} \int \sqrt{\left(x^{2}+2 \cdot x \cdot \frac{2}{3}+\frac{4}{9}\right)-\frac{4}{9}+\frac{1}{3}} d x$
$I=\sqrt{3} \int \sqrt{\left(x+\frac{2}{3}\right)^{2}-\left(\frac{1}{3}\right)^{2}} d x$
$I=\sqrt{3} \int \sqrt{\left(x+\frac{2}{3}\right)^{2}-\left(\frac{1}{3}\right)^{2}} d x$
$I=\sqrt{3}\left[\frac{\left(x+\frac{2}{3}\right)}{2} \sqrt{\left(x+\frac{2}{3}\right)^{2}-\left(\frac{1}{3}\right)^{2}}-\frac{1}{9 \times 2} \log \left(x+\frac{2}{3}+\sqrt{(x+\frac{2}{3})^{2}-\left(\frac{1}{3}\right)^{2}}\right)\right]$
$I=\sqrt{3}\left[\frac{3 x+2}{6} \sqrt{\left(x+\frac{2}{3}\right)^{2}-\left(\frac{1}{3}\right)^{2}}-\frac{1}{18} \log \left(x+\frac{2}{3}+\sqrt{\left(x+\frac{2}{3}\right)^{2}-\left(\frac{1}{3}\right)^{2}}\right)\right.$
$I=\sqrt{3}\left[\frac{3 x+2}{6} \sqrt{\frac{3 x^{2}+4 x+1}{3}}-\frac{1}{18} \log \left(x+\frac{2}{3} \sqrt\frac{3 x^{2}+4 x+1}{2}\right)+c\right.$
$I=\sqrt{3}\left[\frac{3 x+2}{6} \frac{\sqrt{3 x^{2}+4 x+1}}{\sqrt{3}}-\frac{1}{18} \log \left(x+\frac{2}{3} \sqrt\frac{3 x^{2}+9 x+1}{3}\right)\right.$
$I=\frac{3 x+2}{6} \sqrt{3 x^{2}+(x+1)}-\frac{\sqrt{3}}{18} \log \left(x+\frac{2}{3}+\sqrt{\left.\frac{3 x^{2}+(x+1}{3}\right.}\right)+c$
Question 4
$\int \frac{d x}{\sqrt{x^{4}-9 x+8}}$ gives incorrect answer$\int \sqrt{x^{2}-3 x+2} d x$
Sol :
$I=\int \sqrt{x^{2}-3 x+2} d x$
$=\int \sqrt{\left(x^{2}-2 .x \cdot \frac{3}{2}+\frac{9}{4}\right)- \frac{9}{4}+2} d x$
$I=\int \sqrt{\left(x^{2}-2 \cdot x.\frac{3}{2}+\frac{9}{4}\right)-\frac{1}{4}} d x$
$=\int \sqrt{\left(x-\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^2} d x$
$I=\frac{(x-\frac{
3}{2})}{2} \sqrt{(x-\frac{3}{2})^{2}-(\frac{1}{2})^{2}}-\frac{\left(\frac{1}{2}\right)^{2}}{1} \log (x-\frac{3}{2}+\sqrt{(x-\frac{3}{2})^{2}-\frac{1}{2}})^{2}$
$I=\frac{2 x-3}{2 x e} \sqrt{x^{3}-3x+2}-\frac{1}{8} \log \left(x-\frac{3}{2}+\sqrt{x^{2}-3 x+8}\right)+c$
$I=\frac{2 x-3}{4} \sqrt{x^{2}-3 x+2}-\frac{1}{8} \log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c$
Question 5
$\int \frac{d x}{\sqrt{10-8 x-2 x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{10-8 x-2 x^{2}}}$
$=\int \frac{d x}{\sqrt{2\left(5-4 x-x^{2}\right)}}$
$=\frac{1}{\sqrt2}\int \frac{d x}{\sqrt{5-4 x-x^{2}}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{5-4 x-x^{2}-4+4}}$
$=\frac{1}{12} \int \frac{d x}{\sqrt{9-\left(x^{2}+4 x+4\right)}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{3^{2}-(x+2)^{2}}}$
$\because\left[\int\frac{d x}{\sqrt{a^{2}-x^{2}}}=sin^{-1}\frac{x}{a}\right]$
$I=\frac{1}{\sqrt{2}} \sin ^{-1}\left(\frac{x+2}{3}\right)+c$
Question 6
$\int \frac{d x}{\sqrt{4-2 x-2 x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{4-2 x-2 x^{2}}}=\int \frac{d x}{\sqrt{2\left(2-x-x^{2}\right)}}$
$=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{2-x-x^{2}}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{2-\left(x^{2}+x\right)}}$
$=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{2-\left(x^{2}+2 \cdot x \cdot \frac{1}{2}+\frac{1}{4}-\frac{1}{4}\right)}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{2-\left(x^{2}+2 \cdot x+\frac{1}{2}+\frac{1}{4}\right)+\frac{1}{4}}}$
$=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{\frac{9}{4}-\left(x+\frac{1}{2}\right)^{2}}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{\left(\frac{3}{2}\right)^{2}-\left(x+\frac{1}{2}\right)^{2}}}$
$I=\frac{1}{\sqrt{2}} \sin ^{-1} \frac{\left(x+\frac{1}{2}\right)}{\frac{3}{2}}+c$
$=\frac{1}{\sqrt{2}} \sin ^{-1}\frac{(2 x+1)}{2 \times \frac{3}{2}}+c$
$I=\frac{1}{\sqrt{2}} \sin ^{-1} \frac{(2 x+1)}{3}+c$
Question 7
$\int \frac{d x}{\sqrt{16-2 x-2 x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{16-2 x-2 x^{2}}}=\int \frac{d x}{\sqrt{2\left(8-x-x^{2}\right)}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{8-\left(x^{2}+x\right)}}$
$=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{8-\left(x^{2}+2 \cdot x \cdot \frac{1}{2}+\frac{1}{4}-\frac{1}{4}\right)}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{8-\left(x^{2}+2 \cdot x \cdot \frac{1}{2}+\frac{1}{4}\right)+\frac{1}{4}}}$
$=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{\frac{33}{4}-\left(x+\frac{1}{2}\right)^{2}}}$
$I=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{\left(\frac{\sqrt{33} }{2}\right)^{2}-\left(x+\frac{1}{2}\right)^{2}}}$
$I=\frac{1}{\sqrt{2}} \sin ^{-1} \frac{\left(x+\frac{1}{2}\right)}{\frac{\sqrt{33}}{2}}+c$
$=\frac{1}{\sqrt{2}} \sin ^{-1} \frac{(2 x+1)}{2 \times \frac{\sqrt{33}}{2}}+c$
$I=\frac{1}{\sqrt{2}} \sin ^{-1} \frac{\left(2 x^{2}+1\right)}{\sqrt{33} }+c$
Question 8
$\int \frac{d x}{\sqrt{x^{2}+2 x+2}}$Sol :
$I=\int \frac{d x}{\sqrt{x^{2}+2 x+2}}=\int \frac{d x}{\sqrt{x^{2}+2 \cdot x \cdot 1+1-1+2}}$
$I=\int \frac{d x}{\sqrt{\left(x^{2}+2 x+1\right)+1}}$
$=\int \frac{d x}{\sqrt{(x+1)^{2}+(1)^{2}}}$
$I=\log |(x+1)+\sqrt{(x+1)^{2}+1}|+c$
$I=\log |(x+1)+\sqrt{x^{2}+2 x+2}|+c$
Question 9
$\int \frac{d x}{\sqrt{7-6 x-x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{7-6 x-x^{2}}}=\int \frac{d x}{\sqrt{7-\left(x^{2}+6 x\right)}}$
$I=\int \frac{d x}{\sqrt{7-\left(x^{2}+2 \cdot x \cdot 3+9-9\right)}}$
$=\int \frac{d x}{\sqrt{7-\left(x^{2}+2 \cdot x \cdot 3+9\right)+9}}$
$I=\int \frac{d x}{\sqrt{16-\left(x^{2}+2 \cdot x \cdot 3+9\right)}}$
$=\int \frac{d x}{\sqrt{4^{2}-(x+3)^{2}}}$
$I=\sin ^{-1} \frac{(x+3)}{4}+c$
Question 10
$\int \frac{d x}{\sqrt{5 x^{2}-2 x}}$Sol :
$I=\int \frac{d x}{\sqrt{5 x^{2}-2 x}}
$$=\int \frac{d x}{\sqrt{5\left(x^{2}-2 \cdot x \cdot \frac{1}{5}+\frac{1}{25}-\frac{1}{25}\right)}}$
$=\frac{1}{\sqrt{5}} \int \frac{d x}{\sqrt{\left(x^{2}-2 \cdot x \cdot \frac{1}{5}+\frac{1}{25}\right)-\frac{1}{25}}}$
$=\frac{1}{\sqrt{5}} \int \frac{d x}{\sqrt{\left(x-\frac{1}{5}\right)^{2}-\left(\frac{1}{5}\right)^{2}}}$
$I=\frac{1}{\sqrt{5}} \log \left|\left(x-\frac{1}{5}\right)+\sqrt{\left(x-\frac{1}{5}\right)^{2}-\left(\frac{1}{5}\right)^{2}}\right|+c$
$I=\frac{1}{\sqrt{5}} \log \left|\left(x-\frac{1}{5}\right)+\sqrt{x^{2}+\frac{1}{25}-2 \cdot x \cdot \frac{1}{5}-\frac{1}{25}}\right|+c$
$I=\frac{1}{\sqrt{5}} \log \left|\left(x-\frac{1}{5}\right)+\sqrt{x^{2}-\frac{2 x}{5}}\right|+c$
Question 11
$\int \frac{d x}{\sqrt{8+3 x-x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{8+3 x-x^{2}}}$
$=\int \frac{d x}{\sqrt{8-\left(x^{2}-3 x\right)}}$
$I=\int \frac{d x}{\sqrt{8-\left(x^{2}-2 \cdot x \cdot \frac{3}{2}+\frac{9}{4}-\frac{9}{4}\right)}}$
$=\int \frac{d x}{\sqrt{8-\left(x^{2}-2 x \cdot \frac{3}{2}+\frac{9}{4}\right)}}$
$I=\int \frac{d x}{\sqrt{\frac{41}{4}-\left(x-\frac{3}{2}\right)^{2}}}$
$=\int \frac{d x}{\sqrt{\left(\frac{\sqrt{41}}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}}}$
$I=\sin ^{-1} \frac{\left(x-\frac{3}{2}\right)}{\frac{\sqrt{41}}{2}}+c$
$=\sin ^{-1} \frac{(2 x-3)}{2 \times \frac{\sqrt{41}}{2}}+c$
$I=\sin ^{-1}\frac{\left(2 x-3\right)}{\sqrt{41}}+c$
Question 12
$\int \frac{d x}{\sqrt{2 x-x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{2 x-x^{2}}}=\int \frac{d x}{\sqrt{2 x-x^{2}-1+1}}$
$\int \frac{d x}{\sqrt{1-\left(x^{2}-2 x+1\right)}}$
$I=\int \frac{d x}{\sqrt{1-(x-1)^{2}}}$
$=\sin ^{-1} \frac{(x-1)}{1}+c$
$=\sin ^{-1}(x-1)+c$
Question 13
(i) $\int \frac{d x}{\sqrt{5-4 x+x^{2}}}$Sol :
$I=\int \frac{d x}{\sqrt{5-4 x+x^{2}}}$
$=\int \frac{d x}{\sqrt{x^{2}-2 \cdot x \cdot 2+4-4+5}}$
$I=\int \frac{d x}{\sqrt{1+\left(x^{2}-2 \cdot x \cdot 2+4\right)}}$
$=\int \frac{d x}{\sqrt{1^{2}+(x-2)^{2}}}$
$I=\log |(x-2)+\sqrt{(x-2)^{2}+1}|+c$
$I=\log |(x-2)+\sqrt{x^{2}-4 x+5}|+c$
(ii) $\int \frac{d x}{\sqrt{(2-x)^{2}+1}}$
Sol :
$I=\int \frac{d x}{\sqrt{(2-x)^{2}+1}}$
$=\int \frac{d x}{\sqrt{4+x^{2}-4 x+1}}$
$=\int \frac{d x}{\sqrt{x^{2}-4 x+5}}$
$I=\log |(x-2)+\sqrt{x^{2}-4 x+5}|+c$
Question 14
(i) $\int \frac{d x}{\sqrt{(x-1)(x-2)}}$Sol :
$I=\int \frac{d x}{\sqrt{(x-1)(x-2)}}$
$=\int \frac{d x}{\sqrt{x^{2}-2 x-x+2}}$
$=\int \frac{d x}{\sqrt{x^{2}-3 x+2}}$
$I=\int \frac{d x}{\sqrt{x^{2}-2 \cdot x \cdot \frac{3}{2}+\frac{9}{4}-\frac{9}{4}+2}}$
$=\int \frac{d x}{\sqrt{\left(x^{2}-2 x \cdot 3+\frac{9}{4}\right)-\frac{1}{4}}}$
$I=\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}}$
$=\log \left|\left(x-\frac{3}{2}\right)+\sqrt{\left(x-\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}\right|+c$
$I=\log \left|\left(\frac{2 x-3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c$
(ii) $\int \frac{d x}{\sqrt{(x-1)(2-x)}}$
Sol :
$I=\int \frac{d x}{\sqrt{(x-1)(2-x)}}$
$=\int \frac{d x}{\sqrt{2 x-x^{2}-2+x}}$
$=\int \frac{d x}{\sqrt{3 x-x^{2}-2}}$
$I=\int \frac{d x}{\sqrt{1-x^{2}+3 x-2-1}}$
$I=\int \frac{d x}{\sqrt{1-\left(x^{2}-3 x+3\right)}}$
$=\int \frac{d x}{\left.\sqrt{1-\left(x^{2}-2 \cdot x \cdot \frac{3}{2}\right.}+\frac{9}{4}-\frac{9}{4}+3\right)}$
$I=\int \frac{d x}{\sqrt{1-\left(x^{2}-2 \cdot x \cdot \frac{3}{2}+\frac{9}{4}\right)+\frac{9}{4}-3}}$
$I=\int \frac{d x}{\sqrt{\frac{1}{4}-\left(x-\frac{3}{2}\right)^{2}}}$
$=\int \frac{d x}{\sqrt{\left(\frac{1}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}}}$
$I=\sin ^{-1} \frac{\left(x-\frac{3}{2}\right)}{\frac{1}{2}}+c$
$=\sin ^{-1} \frac{(2 x-3)}{2 \times \frac{1}{2}}+c$
$=\sin ^{-1}(2 x-3)+c$
Question 15
$\int \frac{d x}{\sqrt{(x-a)(x-b)}}$Sol :
$I=\int \frac{d x}{\sqrt{(x-a)(x-b)}}=\int \frac{d x}{\sqrt{x^{2}-b x-a x+a b}}$
$I=\int \frac{d x}{\sqrt{x^{2}-x(a+b)+a b}}$
$=\int \frac{d x}{\sqrt{x^{2}-2 \cdot x \cdot\left(\frac{a+b}{2}\right)+\left(\frac{a+b}{2}\right)^{2}-\left(\frac{a+b}{2}\right)^{2}+a b}}$
$I=\int \frac{d x}{\sqrt{\left(x^{2}-2 \cdot x \cdot\left(\frac{a+b}{2}\right)+\left(\frac{a+b}{2}\right)\right)^{2}-\left(\frac{a+b}{2}\right)^{2}+a b}}$
$I=\int \frac{d x}{\sqrt{\left(x-\frac{(a+b)}{2}\right)^{2}-\frac{\left(a^{2}+b^{2}+2 a b\right)}{4}+a b}}$
$I=\int \frac{d x}{\sqrt{\left(x-\frac{(a+b)}{2}\right)^{2}-\left(\frac{(a-b)}{2}\right)^{2}}}$
$\left.I=\log \bigg|\left(x-\frac{(a+b)}{2}\right)+\sqrt{\left(x-\left(\frac{a+b}{2}\right)^{2}-\left(\frac{a-b}{2}\right)^{2}\right.}\right|+c$
$I=\log \left|\frac{2 x-a-b}{2}+\sqrt{(x-a)(x-b)}\right|+c$
Question 16
(i) $\int \frac{d x}{x^{2}-16}$Sol :
$I=\int \frac{d x}{x^{2}-16}=\int \frac{d x}{(x)^{2}-(4)^{2}}$
$I=\frac{1}{2 \times 4} \log \left|\frac{x-4}{x+4}\right|+c$
$=\frac{1}{8} \log \left|\frac{x-4}{x+4}\right|+c$
(ii) $\int \frac{d x}{x\left(x^{5}+3\right)}$
Sol :
$I=\int \frac{d x}{x\left(x^{5}+3\right)}$
Let $x^{5}=2 \Rightarrow x=z^{\frac{1}{5}}$
then $5 x^{4} d x=d z \Rightarrow d x=\frac{d z}{5 x^{4}}=\frac{1}{5} \cdot \frac{1}{z^{\frac{7}{3}}} d z$
Now , $I=\int \frac{d x}{x\left(x^{5}+3\right)}$
$=\int \frac{1 \times 1}{5 \cdot z ^{\frac{4}{5}} \cdot z^{\frac{1}{5}}(z+3)}$
$I=\frac{1}{5} \int \frac{d z}{z(z+3)}$
$=\frac{1}{5} \int \frac{d z}{z^{2}+3z}$
$I=\frac{1}{5} \int \frac{d z}{\left(z^{2}+2 \cdot 2 \cdot \frac{3}{2}+\frac{9}{4}\right)-\frac{9}{4}}$
$=\frac{1}{5} \int \frac{dz}{\left(z+\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}$
$I=\frac{1}{5} \cdot \frac{1}{2 \times \frac{3}{2}} \log \left|\frac{z+\frac{3}{2}-\frac{3}{2}}{z+\frac{3}{2}+\frac{3}{2}}\right|+c$
$I=\frac{1}{15} \log \left|\frac{z}{z+3}\right|+c$
$=\frac{1}{15} \log \left|\frac{x^{5}}{x^{5}+3}\right|+c$
Question 17
$\int \frac{d x}{3 x^{2}+13 x-10}$Sol :
$I=\int \frac{d x}{3 x^{2}+13 x-10}$
$=\int \frac{d x}{3\left(x^{2}+\frac{13}{3} x-\frac{10}{3}\right)}$
$I=\frac{1}{3} \int \frac{d x}{\left(x^{2}+2 \cdot x \cdot \frac{13}{6}+\frac{169}{36}-\frac{169}{36}-\frac{10}{3}\right)}$
$I=\frac{1}{3} \int \frac{d x}{\left(x+\frac{13}{6}\right)^{2}-\frac{289}{36}}$
$=\frac{1}{3} \int \frac{d x}{\left(x+\frac{13}{6}\right)^{2}-\left(\frac{17}{6}\right)^{2}}$
$I=\frac{1}{3} \times \frac{1}{2 \times \frac{17}{6}} \log \left|\frac{x+\frac{13}{6}-\frac{17}{6}}{x+\frac{13}{6}+\frac{17}{6}}\right|+c$
$I=\frac{1}{17} \log \left|\frac{x-\frac{4}{6}}{x+5}\right|+c$
$=\frac{1}{17} \log \left|\frac{3 x-2}{3(x+5)}\right|+c$
$I=\frac{1}{17}[\log (3 x-2)-\log [3(x+5)]]+c$
$I=\frac{1}{17}[\log (3 x-2)-\{\log 3+\log (x+5)\}]+c$
$I=\frac{1}{17}[\log (3 x-2)-\log 3-\log (x+5)]+c$
$I=\frac{1}{17} \log (3 x-2)-\frac{1}{17} \log 3-\frac{1}{17} \log (x+5)+c$
$I=\frac{1}{17} \log (3 x-2)-\frac{1}{17} \log (x+5)+c$
$I=\frac{1}{17}[\log (3 x-2)-\log (x+5)]+c$
$I=\frac{1}{17} \log \left|\frac{3 x-2}{x+5}\right|+c$
Question 18
$\int \frac{d x}{x^{2}-6 x+13}$Sol :
$I=\int \frac{d x}{x^{2}-6 x+13}$
$=\int \frac{d x}{x^{2}-2 \cdot x \cdot 3+9-9+13}$
$I=\int \frac{d x}{\left(x^{2}-2 \cdot x \cdot 3+9\right)+4}$
$=\int \frac{d x}{(x-3)^{2}+(2)^{2}}$
$I=\frac{1}{2} \tan ^{-1} \frac{(x-3)}{2}+c$
Question 19
(i) $\int \sqrt{4-x^{2}} d x$Sol :
$I=\int \sqrt{4-x^{2}} d x=\int \sqrt{(2)^{2}-(x)^{2}} d x$
$I=\frac{x}{2} \cdot \sqrt{4-x^{2}}+\frac{4}{2} \sin^{-1} \frac{x}{2}+c$
$I=\frac{x \sqrt{4-x^{2}}}{2}+2 \sin ^{-1} \frac{x}{2}+c$
(ii) $\int \sqrt{1-4 x^{2}}$
Sol :
$I=\int \sqrt{1-4 x^{2}} d x$
$=\int \sqrt{(1)^{2}-(2 x)^{2}} d x$
$I=\frac{2 x}{2} \sqrt{1-4 x^{2}}+\frac{1}{2} \sin ^{-1}\left(\frac{2 x}{1}\right)+c$
$I=x \sqrt{1-4 x^{2}}+\frac{1}{2} \sin ^{-1}(2 x)+c$
$I=\frac{x \sqrt{1-4 x^{2}}}{2}+\frac{1}{4} \sin ^{-1}(2 x)+c$
Question 20
(i) $\int \sqrt{3-2 x- x^{2}}$Sol :
$I=\int \sqrt{3-2 x- x^{2}}$
$\int \sqrt{3-\left(x^{2}+2 x\right)}$
$I=\int \sqrt{3-\left(x^{2}+2. x.1+1-1\right)}$
$=\int \sqrt{3-\left(x^{2}+2 x+1\right)+1}$
$I=\int \sqrt{4-\left(x^{2}+2 x+1\right)}$
$=\int \sqrt{(2)^{2}-(x+1)^{2}}$
$I=\frac{(x+1) \cdot \sqrt{4-(x+1)^{2}}}{8}+\frac{4}{2} \sin ^{-1} \frac{(x+1)}{2}+c$
$I=\frac{(x+1)}{2} \sqrt{3-2 x-x^{2}}+2 \sin ^{-1}\left(\frac{x+1}{2}\right)+c$
(ii) $\int \sqrt{1-4 x-x^{2}} d x$
Sol :
$I=\int \sqrt{1-4 x-x^{2}} d x$
$=\int \sqrt{1-\left(x^{2}+4 x\right)} d x$
$I=\int \sqrt{1-\left(x^{2}+2 x \cdot 2+4-4\right)} d x$
$=\int \sqrt{1-\left(x^{2}+2 x \cdot 2+4\right)+4}$
$I=\sqrt{5-\left(x^{2}+2.x .2+4\right)}$
$=\sqrt{(\sqrt{5})^{2}-(x+9)^{2}} d x$
$I=\frac{(x+2)}{2} \cdot \sqrt{5-(x+2)^{2}}+\frac{5}{2} \sin ^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+c$
$I=\frac{(x+2)}{2} \sqrt{1-4 x-x^{2}}+\frac{5}{2} \sin ^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+c$
Question 21
$\int \sqrt{1+3 x-x^{2}} d x$Sol :
$I=\int \sqrt{1+3 x-x^{2}} d x$
$=\int \sqrt{1-\left(x^{2}-3 x\right)} d x$
$I=\int \sqrt{1-\left(x^{2}-2 \cdot x \cdot \frac{3}{2}+\frac{9}{4}-\frac{9}{4}\right)} d x$\
$I=\int \sqrt{1-\left(x^{2}-2 x \frac{3}{2}+\frac{9}{4}\right)+\frac{9}{4}}$
$=\int \sqrt{\frac{13}{4}-\left(x-\frac{3}{2}\right)^{2}} d x$
$I=\int \sqrt{\left(\frac{\sqrt{13}}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}} d x$
$I=\frac{\left(x-\frac{3}{2}\right)}{2} \sqrt{\left(\frac{\sqrt{3}}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}}+\frac{13}{4 \times 2} \sin ^{-1} \frac{\left(x-\frac{3}{2}\right)}{\sqrt{3}}+c$
$I=\frac{2 x-3}{4} \cdot \sqrt{1+3 x-x^{2}}+\frac{13}{8} \sin ^{-1} \frac{(2 x-3)}{\frac{\sqrt{13}}{2} \times 2}+c$
$I=\frac{(2 x-3)}{4} \sqrt{1+3 x-x^{2}}+\frac{13}{8} \sin ^{-1} \frac{(2 x-3)}{\sqrt{13}}+c$
Question 22
$\int \sqrt{x^{2}+4 x-5} d x$Sol :
$I=\int \sqrt{x^{2}+4 x-5} d x$
$I=\int \sqrt{\left(x^{2}+2 \cdot x \cdot 2+4\right)-9} d x$
$=\int \sqrt{(x+2)^{2}-(3)^{2}}$
$I=\frac{(x+2)}{2} \sqrt{(x+2)^{2}-(3)^{2}}-\frac{9}{2} \log |(x+2)+\sqrt{(x+2)^{2}-(3)^{2}}|+c$
$I=\frac{x+2}{2} \sqrt{x^{2}+4 x-5}-\frac{9}{2} \log |x+2+\sqrt{x^{2}+4 x-5}|+c$
Question 23
$\int \sqrt{x^{2}+4 x+1} d x$Sol :
$I=\int \sqrt{x^{2}+4 x+1} d x$
$=\int \sqrt{x^{2}+2 \cdot x \cdot 2+4-4+1} d x$
$I=\int \sqrt{\left(x^{2}+2 \cdot x \cdot 2+4\right)-3} d x$
$=\int \sqrt{(x+2)^{2}-(\sqrt{3})^{2}} d x$
$I=\frac{x+2}{2} \sqrt{(x+2)^{2}-(\sqrt{3})^{2}}-\frac{3}{2} \log |x+2+\sqrt{(x+2)^{2}-(\sqrt{3})^{2}}|+c$
I=\frac{(x+2)}{2} \sqrt{x^{2}+4 x+1}-\frac{3}{2} \log \bigg|x+2+\sqrt{x^{2}+ 4x+1}\bigg|+c
Question 24
(i) $\int \sqrt{x^{2}+3 x} d x$Sol :
$I=\int \sqrt{x^{2}+3 x} d x$
$=\int \sqrt{\left(x^{2}+2 \cdot x \cdot \frac{3}{2}+\frac{9}{4}-\frac{9}{4}\right)} d x$
$I=\int \sqrt{\left(x+\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}$
$I=\frac{\left(x+\frac{3}{2}\right)}{2} \cdot \sqrt{\left(x+\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}-\frac{9}{4 \times 2} \log |\left(x+\frac{3}{2}\right)+\sqrt{\left(x+\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}$
$I=\frac{2 x+3}{4} \sqrt{x^{2}+3 x}-\frac{9}{8} \log \left|x+\frac{3}{2}+\sqrt{x^{2}+3 x}\right|+c$
(ii) $\int \sqrt{x^{2}-8 x+7} d x$
Sol :
$I=\int \sqrt{x^{2}-8 x+7} d x$
$=\int \sqrt{x^{2}-2 \cdot x \cdot 4+16-16+7} d x$
$I=\int \sqrt{\left(x^{2}-2 \cdot x \cdot 4+16\right)-9}dx$
$=\int \sqrt{(x-4)^{2}-(3)^{2}} d x$
$I=\frac{(x-4)}{2} \sqrt{(x-4)^{2}-(3)^{2}}-\frac{9}{2} \log \bigg|x-4+\sqrt{(x-4)^{2}-(3)^{2}}\bigg|+c$
$I=\frac{x-4}{2} \sqrt{x^{2}-8 x+7}-\frac{9}{2} \log \bigg|x-4+\sqrt{x^{2}-8 x+7}\bigg|+c$
Question 25
(i) $\int \sqrt{1+x^{2}} d x$Sol :
$I=\int \sqrt{1+x^{2}} d x=\int \sqrt{(1)^{2}+(x)^{2}} d x$
$I=\frac{x}{2} \sqrt{x^{2}+1}+\frac{1}{2} \log \big|x+\sqrt{x^{2}+1}\big|+c$
$I=\frac{x \sqrt{1+x^{2}}}{2}+\frac{1}{2} \log \big|x+\sqrt{1+x^{2}}\big|+c$
(ii) $\int \sqrt{1+\frac{x^{2}}{9}} d x$
Sol :
$I=\int \sqrt{1+\frac{x^{2}}{9} d x}$
$=\int \sqrt{\frac{9+x^{2}}{9} d x}$
$=\int \frac{\sqrt{9+x^{2}}}{3} d x$
$I=\frac{1}{3} \int \sqrt{x^{2}+9} d x$
$=\frac{1}{3} \int \sqrt{(x)^{2}+(3)^{2}} d x$
$I=\frac{1}{3}\left[\frac{x}{2} \sqrt{x^{2}+9}+\frac{9}{2} \log \big| x+\sqrt{x^{2}+9}\big|\right] +c$
$I=\frac{x}{6} \sqrt{x^{2}+9}+\frac{3}{2} \log \big|x+\sqrt{x^{2}+9}\big|+c$
Question 26
$\int \sqrt{x^{2}+4 x+6} d x$Sol :
$I=\int \sqrt{x^{2}+4 x+6} d x$
$=\int \sqrt{x^{2}+2 \cdot x \cdot 2+4-4+6} d x$
$I=\int \sqrt{\left(x^{2}+2 \cdot x \cdot 2+4\right)+2} d x$
$=\int \sqrt{(x+2)^{2}+(\sqrt{2})^{2}} d x$
$I=\frac{(x+2)}{2} \sqrt{(x+2)^{2}+(\sqrt{2})^{2}}+\frac{2}{2} \log |x+2+\sqrt{(x+2)^{2}+(\sqrt{2})^{2}}|+c$
$I=\frac{x+2}{2} \sqrt{x^{2}+4 x+6}+\log \big|x+2+\sqrt{x^{2}+4 x+6}\big|+c$
Question 27
$\int \sqrt{x^{2}+2 x+5} d x$Sol :
$I=\int \sqrt{x^{2}+2 x+5} d x$
$I=\int \sqrt{\left(x^{2}+2 .x. 1+1\right)-1+5} d x$
$I=\int \sqrt{(x+1)^{2}+(2)^{2}} d x$
$I=\frac{(x+1)}{2} \cdot \sqrt{(x+1)^{2}+(2)^{2}}+\frac{4}{2} \log \big|(x+1)+\sqrt{(x+1)^{2}+(2)^{2}}\big|+c$
$I=\frac{x+1}{2} \cdot \sqrt{x^{2}+2 x+5}+2 \log \big|x+1+\sqrt{x^{2}+2 x+5}\big|+c$
Question 28
$\int \frac{x+3}{x^{2}-2 x-5} d x$Sol :
$I=\int \frac{x+3}{x^{2}-2 x-5} d x$
Let $x+3=A \cdot \frac{d}{d x}(x-2 x-5)+B$
x+3=A(2x-2)+B⇒2A.x-2A+B
Equating the coefficient of the similar power of x
2A=1 and 3=-2A+B
$A=\frac{1}{2}$ and B=3+2A$=3+2 \times \frac{1}{2}$
=3+1=4
Now , $I=\int \frac{x+3}{x^{2}-2 x-5} d x$
$=\int \frac{\frac{1}{2}(2 x-8)+4}{x^{2}-8 x-5} d x$
$I=\frac{1}{2} \int \frac{2 x-2}{x^{2}-2 x-5} d x+4 \int \frac{1}{x^{2}-2 x-5} d x$
$I=\frac{1}{2} \int \frac{1}{z}\times d z+4\int \frac{1}{\left(x^{2}-2 .x .1+1\right)-1-5)} d x$
$I=\frac{1}{2} \log |z|+4 \int \frac{d x}{(x-1)^{2}-(\sqrt{6})^2}$
$I=\frac{1}{2} \log \left|x^{2}-2 x-5\right|+4\left[\frac{1}{2\sqrt6} \log \left|\frac{x-1-\sqrt{6}}{x-1+\sqrt{8}}\right|\right]+c$
$I=\frac{1}{2} \log \left|x^{2}-2 x-5\right|+\frac{2}{\sqrt{6}} \log \left|\frac{x-1-\sqrt{6}}{x-1+ \sqrt{6}}\right|+c$
Question 29
$\int \frac{5 x-2}{1+2 x+3 x^{2}} d x$Sol :
$I=\int \frac{5 x-2}{1+2 x+3 x^{2}} d x$
Let 5x-2=$=A \frac{d}{d x}\left(3 x^{2}+2 x+1\right)+B$
5x-2=A(6x+2)+B
=6Ax+2A+B
Equating the coefficient of similar power of x
6A=5 and 2A+B=-2
$A=\frac{5}{6}$ and B=-2-2A=-2-2×5=-12
$B=-\frac{11}{3}$
Now , $I=\int \frac{5 x-2}{3 x^{2}+2 x+1} d x$
$=\int \frac{\frac{5}{6}(6 x+2)-\frac{11}{3}}{3 x^{2}+2 x+1} d x$
$I=\frac{5}{6} \int \frac{6 x+2}{3 x^{2}+2 x+1} d x-\frac{11}{3} \int \frac{d x}{3 x^{2}+2 x+1}$
$I=\frac{5}{6} \int \frac{d z}{z}-\frac{11}{3} \int \frac{d x}{3\left(x^{2}+2 \cdot x \frac{1}{3}+\frac{1}{3}\right)}$
$I=\frac{5}{6} \log |z|-\frac{11}{3 \times{3}} \int \frac{d x}{\left(x^{2}+2 \cdot x \cdot \frac{1}{3}+\frac{1}{9}-\frac{1}{9}+\frac{1}{3}\right)}$
$I=\frac{9}{6} \log \left|3 x^{2}+2 x+1\right|-\frac{11}{9} \int \frac{d x}{\left(x+\frac{1}{3}\right)^{2}+\left(\frac{\sqrt2}{9}\right)^2}$
$I=\frac{5}{6} \log \left|3 x^{2}+2 x+1\right|-\frac{11}{9} \times \frac{1}{\frac{\sqrt{2}}{3}} \tan ^{-1} \dfrac{\left(x+\frac{1}{3}\right)}{\frac{\sqrt2}{3}}+c$
$I=\frac{5}{6} \log \left|3 x^{2}+2 x+1\right|-\frac{11}{3 \sqrt{2}} \tan ^{-1} \frac{(3 x+1)}{3\times \frac{\sqrt2}{3}}+c$
$I=\frac{5}{6} \log \left|3 x^{8}+2 x+1\right|-\frac{11}{3 \sqrt{2}} \tan ^{-1} \frac{(3 x+1)}{\sqrt{2}}+C$
Question 30
$\int \frac{x+2}{2 x^{2}+6 x+5} d x$Sol :
$I=\int \frac{x+2}{2 x^{2}+6 x+5} d x$
Let $x+2=A=\frac{d}{d x}(2 x^2+6 x+5)+B$
x+2=A(4x+6)+B
=4Ax+6A+B
Equating the coefficient of similar power of x
4A=1
$A=\frac{1}{4}$
and 6A+B=2
B=2-6A$=2-6 \times \frac{1}{4}=2-\frac{3}{2}$
$B=\frac{1}{2}$
Now , $I=\int \frac{x+2}{2 x^{2}+6 x+5} d x$
$=\int \frac{\frac{1}{4}(4 x+6)+\frac{1}{2}}{2 x^{2}+6 x+5} d x$
$I=\frac{1}{4} \int \frac{4 x+6}{2 x^{2}+6 x+5} d x+\frac{1}{2} \int \frac{d x}{2 x^{2}+6 x+5}$
$I=\frac{1}{4} \int \frac{d z}{z}+\frac{1}{2} \int \frac{d x}{2\left(x^{2}+3 x+\frac{5}{2}\right)}$
$I=\frac{1}{4} \int \frac{d z}{2}+\frac{1}{2 \times 2} \int \frac{d x}{\left(x^{2}+2 .x.\frac{3}{8}+\frac{9}{4}-\frac{9}{4}+\frac{5}{2}\right)}$
$I=\frac{1}{4} \log z+\frac{1}{4} \int \frac{d x}{\left(x+\frac{3}{2}\right)^{2}+\frac{1}{4}}$
$I=\frac{1}{4} \log z+\frac{1}{4} \int \frac{d x}{\left(x+\frac{3}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}$
$I=\frac{1}{4} \log \left|2 x^{2}+6 x+5\right|+\frac{1}{4} \frac{1}{\frac{1}{2}} \tan ^{-1}\frac{\left(x+\frac{3}{2}\right)}{\frac{1}{2}}+c$
$I=\frac{1}{4} \log \left|2 x^{2}+6 x+5\right|+\frac{2}{4} \tan ^{-1} \frac{(2 x+3)^{\frac{1}{2}}}{2\times \frac{1}{2}}+c$
$I=\frac{1}{4} \log \left|2x^{2}+6 x+5\right| +\frac{1}{2} \tan ^{-1}(2 x+3)+c$
Question 31
$\int \frac{3 x+1}{2 x^{2}-2 x+3} d x$Sol :
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