KC Sinha Solution Class 12 Chapter 19 Indefinite Integrals Exercise 19.1

Exercise 19.1

---

Question 1
∫(x²-5x+7)dx
Sol :
= ∫x²dx-5∫xdx+7∫dx

[∵ ∫xⁿdx=$\dfrac{x^{n+1}}{n+1}$]

=$\dfrac{x^3}{3}-\dfrac{5x^2}{2}+7x+c$

Question 2

∫(ax³-bx²+cx+d)dx
Sol :

=a∫x³dx+b∫x²dx+c∫xdx+d∫dx

=$a\dfrac{x^4}{4}+b\dfrac{x^3}{3}+c\dfrac{x^2}{2}+dx+c$

Question 3

$\displaystyle\int \left(x+\dfrac{1}{x}+2\right)$
Sol :
= ∫xdx+$\int \dfrac{1}{x}dx$+2∫dx

=$\dfrac{x^2}{2}$+log|x|+2x+c

Question 4

∫(x^1/3+2x^1/2+x^3/2)dx
Sol :
=∫x^1/3dx+2∫x^1/2dx+∫x^3/2dx

=$\dfrac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}+2 \times \dfrac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\dfrac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}$+c

=$\dfrac{x^{4/3}}{4/3}+\dfrac{2x^{3/2}}{3/2}+\dfrac{x^{5/2}}{5/2}$+c

=$\dfrac{3}{4}x^{4/3}+\dfrac{2}{3}\times 2 x^{3/2}+\dfrac{2}{5}x^{5/2}$+c

=$\dfrac{3}{4}x^{4/3}+\dfrac{4}{3}x^{3/2}+\dfrac{2}{5}x^{5/2}$+c

Question 5

$\displaystyle\int \dfrac{x^2-2x+3}{x^4}dx$

Sol :

=$\displaystyle\int \dfrac{x^2}{x^4}dx-2\displaystyle\int \dfrac{x}{x^4}dx+3\displaystyle\int \dfrac{1}{x^4}dx$

=$\displaystyle\int \dfrac{1}{x^2}dx-2\displaystyle\int \dfrac{1}{x^3}dx+3\displaystyle\int \dfrac{1}{x^4}dx$

=$\displaystyle\int x^{-2}dx-2\displaystyle\int x^{-3}dx+3\displaystyle\int x^{-4}dx$

=$\dfrac{x^{-2+1}}{-2+1}-2\dfrac{x^{-3+1}}{-3+1}+3\dfrac{x^{-4+1}}{-4+1}$+c

=$\dfrac{x^{-1}}{-1}-2\dfrac{x^{-2}}{-2}+3\dfrac{x^{-3}}{-3}$+c

=$-\dfrac{1}{x}+\dfrac{1}{x^2}-\dfrac{1}{x^3}$+c


Question 6
$\displaystyle\int \left(\sqrt{x}+\dfrac{1}{\sqrt{x}}^2 \right)dx$
Sol:
=$\displaystyle\int \left\{(\sqrt{x})^2+\left(\dfrac{1}{\sqrt{x}}\right)+2 .\sqrt{x}.\dfrac{1}{\sqrt{x}}\}$

=$\displaystyle\int \left(x+\dfrac{1}{x}+2\right)dx$

=$\displaystyle\int xdx+\displaystyle\int \dfrac{1}{x} dx+2\displaystyle\int dx$

=$\dfrac{x^2}{2}$+log|x|+2x+c

Question 7

$\displaystyle\int \dfrac{(x-3)^2}{\sqrt{x}}dx$

Sol:

=$\displaystyle\int \dfrac{x^2+9-2.x.3}{\sqrt{x}}dx$

=$\displaystyle\int \dfrac{x^2}{\sqrt{x}}dx+9\displaystyle\int \dfrac{1}{\sqrt{x}}dx-6\displaystyle\int \dfrac{x}{\sqrt{x}}dx$

=$\displaystyle\int x^{2-\frac{1}{2}}dx+9\displaystyle\int x^{-1/2}dx-6\displaystyle\int \sqrt{x}dx

=$\displaystyle\int x^{3/2}dx+9\displaystyle\int x^{-1/2}dx-6\displaystyle\int x^{1/2}dx

=$\dfrac{x^{5/2}}{5/2}+9\dfrac{x^{1/2}}{1/2}-6\dfrac{x^{3/2}}{3/2}$

=$\dfrac{2}{5}x^{5/2}+18.x^{1/2}-4x^{3/8}+c$

=$\dfrac{2}{5}\sqrt{x}(x^2+45-10x)+c$

=$\dfrac{2}{5}\sqrt{x}(x^2-10x+45)+c$

Question 8

$\displaystyle \int \dfrac{(x^3+1)(x-2)}{x^2-x-2}dx$

Sol :

[(a³+b³)=(a+b)(a²+b²-ab)]

=$\displaystyle \int \dfrac{(x+1)(x^2+1-x)(x-2)}{x^2-x-2}$

=$\displaystyle \int \dfrac{(x+1)(x-2)(x^2+1-x)}{x^2-2x+x-2}$

=$\displaystyle \int \dfrac{(x+1)(x-2)(x^2+1-x)}{x(x-2)+1(x-2)}$

=$\displaystyle \int \dfrac{(x+1)(x-2)(x^2+1-x)}{(x+1)(x-2)}$

=$\displaystyle\int (x^2+1-x)dx$

=$\displaystyle\int x^2dx+1\displaystyle\int dx-\displaystyle\int xdx$

=$\dfrac{x^3}{3}+x-\dfrac{x^2}{2}$+c


Question 9
$\displaystyle\int (ax^2+bx+c)dx$
Sol :
=$\displaystyle\int ax^2dx+\displaystyle\int bxdx+\displaystyle\int cdx$

=$a\displaystyle\int x^2dx+b\displaystyle\int xdx+c\displaystyle\int dx$

=$a.\dfrac{x^3}{3}+b\dfrac{x^2}{2}+cx$+k

Question 10

$\displaystyle\int (3x^3+4x^3)dx$
Sol :
=$\displaystyle\int 3x^2dx+\displaystyle\int 4x^3dx$

=$3 \displaystyle\int x^2dx+4\displaystyle\int x^3dx$

=$3\dfrac{x^3}{3}+4\dfrac{x^4}{4}$+c

=x³+x⁴+c


Question 11
$\displaystyle\int \dfrac{x^3+5x^2-4}{x^2}dx$
Sol :
=$\displaystyle\int \dfrac{x^3}{x^2}dx+5\displaystyle\int \dfrac{x^2}{x^2}dx-4\displaystyle\int \dfrac{1}{x^2}dx$

=$\displaystyle\int x dx+5\displaystyle\int dx-4\displaystyle\int x^{-2}dx$

=$\dfrac{x^2}{2}+5x-4 \times \dfrac{x^{-2+1}}{-2+1}$+c

=$\dfrac{x^2}{2}+5x-4 \times \dfrac{x^{-1}}{-1}$+c

=$\dfrac{x^2}{2}+5x + \dfrac{4}{x}$+c

Question 12
$\int \frac{x^{3}-1}{x^{2}} d x$
Sol :
$=\int \frac{x^{3}}{x^{2}} d x-\int \frac{1}{x^{2}} d x$

$\int x d x-\int x^{-2} d x$

$=\frac{x^{2}}{2}-\frac{x^{-2+1}}{-2+1}+c$

$\frac{x^{2}}{2}-\frac{x^{-1}}{-1}+c$

$=\frac{x^{2}}{2}+\frac{1}{x}+c$

Question 13

$\int x^{2}\left(1-\frac{1}{x^{2}}\right) d x$

Sol:

Given, $\int x^{2}\left(1-\frac{1}{x^{2}}\right) d x$

$\int x^{2} d x-\int x^{2}+\frac{1}{x^{2}} d x$

$=\int x^{2} d x-\int d x$

$=\frac{x^{3}}{3}-x+c$

Question 14
$\int \frac{x^{3}+3 x+4}{\sqrt{x}} d x$
Sol:
$=\int \frac{x^{3}}{\sqrt{x}} d x+3 \int \frac{x}{\sqrt{x}} d x+4 \int \frac{1}{\sqrt{x}} d x$

$=\int \frac{x^{3}}{x^{1 / 2}} d x+3 \int \frac{\sqrt{x} \cdot \sqrt{x}}{\sqrt{x}}+4 \int \frac{1}{x^{1 / 2}} d x$


$=\int x^{3-\frac{1}{2}} d x+3 \int \sqrt{x} d x+4 \int x^{-1 / 2} d x$

$=\int x^{5 / 2} d x+3 \int x^{1 / 2} d x+4 \int x^{-1 / 2} d x$

$=\frac{x^{-\frac{5}{2}+1}}{\frac{5}{2}+1}+3 \cdot \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+4 \cdot \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}$

=\frac{2}{7} \cdot x^{\frac{7}{2}}+3 \frac{x^{3 / 2}}{\frac{3}{2}}+\frac{4 \cdot \frac{x^{1/2}}{\frac{1}{2}}}{\frac{1}{2}}

=$\dfrac{2}{7}.x^{\frac{3}{2}}+2x^{3/2}+8x^{1/2}+c$

Question 15
$\int\left(x^{2 / 3}+1\right) d x$
Sol :
$=\int x^{\frac{2}{3}} d x+\int 1 d x$

$=\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1}+x+c$

$=\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+x+c$

$=\frac{3}{5} \cdot x^{\frac{5}{3}}+x+c$

Question 16

$\int(1-x) \sqrt{x} d x$
Sol :
$=\int x^{\frac{1}{2}} d x-\int x \cdot x^{\frac{1}{2}} d x$

$\left(x^{m} \cdot x^{\pi}=x^{m+n}\right)$

$=\int x^{\frac{1}{2}} d x-\int x^{1+\frac{1}{2}} d x$

$=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{x^{1+\frac{1}{2}+1}}{1+\frac{1}{2}+1}+c$

$=\frac{x^{\frac{3}{2}}}{\frac{2}{3}}-\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c$

$=\frac{3}{2} \cdot x^{3/2}$

Question 17
$\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} d x$
Sol :
$=\int\left(x+\frac{1}{x}-2\right) d x$

$=\int x d x+\int \frac{1}{x} d x-2 \int d x$

=$\frac{x^{2}}{2}+\log |x|-2 x+c$

Question 18
$\int \sqrt{x}\left(3 x^{2}+2 x+3\right) d x$
Sol :
$=\int \sqrt{x} \cdot 3 x^{2} d x+\int \sqrt{x} \cdot 2 x d x+\int \sqrt{x} \cdot 3 d x$

$=3 \int x^{\frac{1}{2}} \cdot x^{2} d x+2 \int x^{\frac{1}{2}} \cdot x d x+3 \int x^{\frac{1}{2}} d x$

$=3 \int x^{\frac{1}{2}+2} d x+2 \int x^{\frac{1}{2}+1} d x+3 \int x^{\frac{1}{2}} d x$

$=3 \int x^{\frac{5}{3}} d x+2 \int x^{\frac{3}{2}} d x+3 \int x^{\frac{1}{2}} d x$

$=3 . \dfrac{x^{\frac{5}{3}+1}}{\frac{5}{3}+1}+2. \dfrac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1} +3.\dfrac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} +c$


$=3 \cdot \frac{x^{\frac{7}{2}}}{\frac{7}{2}}+2 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}}+3 \frac{x^{\frac{3}{2}}}{\frac{3}{2}}$

$=\frac{6}{7} \cdot x^{\frac{7}{2}}+\frac{4}{5} x^{\frac{5}{2}}+2 x^{\frac{3}{2}}+c$


Question 19

$\int \frac{x^{3}-x^{2}+x-1}{x-1} d x$

Sol :

$=\int \frac{x^{2}(x-1)+(x-1)}{x-1} d x$

$=\int \frac{(x+1)\left(x^{2}+1\right)}{x-1} d x$

$=\int\left(x^{2}+1\right) d x$

$=\int x^{2} d x+\int d x$

$=\frac{x^{3}}{3}+x+c$

Question 20
$\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$
Sol :
$=\int x^{\frac{1}{2}} d x+\int x^{-\frac{1}{2}} d x$

$=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+c$

$=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{\frac{2}{2}}}{\frac{1}{2}}+c$

$=\frac{2}{3} \cdot x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+c$

$=\frac{2}{3} x^{\frac{3}{2}}+2 \sqrt{x}+c$

Question 21
$\int\left(4 e^{3 x}+1\right) d x$
Sol :
$=y \int e^{3 x} d x+\int d x$

$=4 \frac{e^{3 x}}{3}+x+c$

$=\frac{4}{3} \cdot e^{3 x}+x+c$


Question 22
$\int\left(x^{\frac{3}{2}}+2 e^{x}-\frac{1}{x}\right) d x$
Sol :
$=\int x^{\frac{3}{2}} d x+2 \int e^{x}-\int \frac{1}{x} d x$

$=\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+2 e^{x}-\log |x|+c$

=$\dfrac{2}{5}.x^{\frac{5}{2}}+2.e^x-log|x|+c$

Question 23

$\int\left(2 x^{2}+e^{x}\right) d x$

Sol :

$=2 \int x^{2} d x+\int e^{x} d x$

$=2 \int x^{2} d x+\int e^{x} d x$

$=2 \cdot \frac{x^{3}}{3}+e^{x}+c$

$=\frac{2}{3} \cdot x^{3}+e^{x}+c$

Question 24
$\int(\sin x+\cos x) d x$
Sol :
$=\int \sin x d x+\int \cos x d x$

$=-\cos x+\sin x+c$

Question 25
$\int \frac{2-3 \sin x}{\cos ^{2} x} d x$
Sol :
$=\int \frac{2}{\cos ^{2} x} d x-3 \int \frac{\sin x}{\cos ^{2} x} d x$

=2 \int \sec ^{2} x d x-3 \int \frac{\sin x}{\cos x \cdot \cos x} d x

$=2 \int \sec ^{2} x d x-3 \int \tan x \cdot \sec x d x$

=2 \tan x-3 \sec x+c

Question 26
$\int \operatorname{cosec} x(\operatorname{cosec} x+\cot x) d x$
Sol :
$=\int \operatorname{cosec} x \cdot \operatorname{cosec} x d x+\int \operatorname{cosec} x \cdot \cot x d x$

$=\int \operatorname{cosec}^{2} x d x+\int \operatorname{cosec} x \cdot \cot x d x$

$=\cot x-\operatorname{cosec} x+c$

$=(-\cot x+\operatorname{cosec} x)+c$

Question 27
$\int \sec x(\sec x+\tan x) d x$
Sol :
$=\int \sec ^{2} x d x+\int \sec x \cdot \tan x d x$

$=\tan x+s e c x+c$

Question 28
$\int \frac{1-\sin x}{\cos ^{2} x} d x$
Sol :
$=\int \frac{1}{\cos^2 x} d x-\int \frac{\sin x}{\cos x.cosx} d x$

$=\int \sec ^{2} x d x-\int \sec x \cdot \tan x d x$

$=\tan x-\sec x+c$

Question 29
$\int\left(\sec ^{2} x+\operatorname{cosec}^{2} x\right) d x$
Sol :
$=\int \sec ^{2} x d x+\int \operatorname{cosec}^{2} x d x$

$=\tan x-\cot x+c$

Question 30
$\int\left(\sin 2 x-4 e^{3 x}\right) d x$
Sol :
$\int \sin 2 x d x-4 \int e^{3 x} d x$

$=\frac{-\cos 2 x}{2}-\frac{4 e^{3 x}}{3}+c$

$=\frac{-\cos 2 x}{2}-\frac{y}{3} \cdot e^{3 x}+c$

Question 31

$\int\left(2 x-3 \cos x+e^{x}\right) d x$

Sol :

$=2 \int x d x-3 \int \cos x d x+\int e^{x} d x$

$=2\times \frac{x^{2}}{2}-3 \sin x+e^{x}+c$

$x^{2}-3 \sin x+e^{x}+c$

Question 32
$\int\left(2 x^{2}-3 \sin x+5 \sqrt{x}\right) d x$
Sol :
$=2 \int x^{2} d x-3 \int \sin x d x+5 \int \sqrt{x} d x$

=$2 \cdot \frac{x^{3}}{3}-3(-\cos x)+5 \cdot \frac{x^{-2}+1}{\frac{1}{2}+1}$+c

=$\frac{2}{3} x^{3}+3 \cos x+5 \cdot \frac{x^{3/2}}{\frac{3}{2}}+c$

=$\frac{2}{3} x^{3}+3 \cos x+\frac{10}{3} \cdot x^{\frac{3}{2}}+c$

Question 33

(i)$\int\left(5 \cos x-4 \sin x+\dfrac{1}{\cos ^{2} x}\right) d x$

Sol :
$=5 \int \cos x d x-4 \int \sin x+\int \sec ^{2} x d x$

=5sinx-4(-cosx)+tanx+c

=5sinx+4cosx+tanx+c

(ii) $\int \frac{\sin ^{2} x-\cos ^{2} x}{1-2 \sin ^{2} x \cdot \cos ^{2} x} d x$

Sol :

$\int \frac{\sin ^{2} x-\cos ^{2} x}{1-2 \sin ^{2} x \cdot \cos ^{2} x} d x$

$\int \frac{\left(\sin ^{4} x\right)^{2}-\left(\cos ^{4} x\right)^{2}}{1-2 \sin ^{2} x \cdot \cos ^{2} x} d x$

$\int \frac{\left(\sin ^{4} x-\cos ^{4} x\right)\left(\sin ^{4} x+\cos ^{4} x\right)}{1-2 \sin ^{2} x \cdot \cos ^{2} x} d x$

$\displaystyle\int\dfrac{\left( \sin ^{2} x\right)^{2}-\left(\cos ^{2} x\right)^{2}\left\{\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cdot \cos ^{2} x \right)}{1-2sin^2x.cos^2x}$

$\displaystyle\int \dfrac{\left(\sin ^{2} x-\cos ^{2} x\right)\left(\sin ^{2} x+\cos ^{2} x\right)\left(1-2 \sin ^{2} x-\cos ^{2} x\right) d x}{1-2sin^2x.cos^2x}$

$=\int\left(\sin ^{2} x-\cos ^{2} x\right) d x$

$=-\int\left(\cos ^{2} x-\sin ^{2} x\right) d x$

$=-\int \cos 2 x d x$

$=-\frac{\sin 2 x}{2}+c$

Question 34
$\displaystyle\int \dfrac{1+2 \sin x}{\cos ^{2} x} d x$
Sol :
$=\int \frac{1}{\cos ^{2} x} d x+2 \int \frac{\sin x}{\cos x \cdot \cos x} d x$

$=\int \sec ^{2} x d x+2 \int \sec x \cdot \tan x d x$

=tanx+2secx+c

Question 35

$\int \frac{5cos^{3} x+7 \sin ^{2} x}{\sin ^{2} x \cdot \cos ^{2} x} d x$

Sol :

=$5 \int \frac{\cos ^{3} x}{\sin^2 x.cos^2x}+7\int \frac{\sin ^{3} x}{\sin ^{2} x \cdot \cos ^{2} x}$

$=5\int \frac{\cos x}{\sin x \cdot \sin x} d x+7\int\dfrac{sinx}{cosx.cosx}dx$

=$5\int \frac{\cos x}{\sin x \cdot \sin x} d x+7 \int \frac{\sin x}{\cos x \cdot \cos x} d x$

=$5 \int \cot x \cdot \operatorname{cosec} x d x+7 \int \tan x \cdot \sec x d x$

=5(-cosecx)+7secx+c

=7secx-5cosecx+c

Question 36

$\int \frac{e^{x} \sin x+\cot x+x \sin x}{\sin x} d x$

Sol :

$\int \frac{e^{x} \sin x}{\sin x} d x+\int \frac{\cot x}{\sin x} d x+\int \frac{x \sin x}{\sin x} d x$

$=\int e^{x} d x+\int \cot x \cdot \operatorname{cosec} x+\int x d x$

$e^{x}-\operatorname{cosec} x+\frac{x^{2}}{2}+c$

Question 37

$\int \sec (11+12 x) \cdot \tan (11+12 x) d x$
Sol :
$=\frac{\sec (11+12 x)}{12}+c$

Question 38

$\int(\cos x+\sin x)^{2} d x$

Sol :

=$\int\left(\cos ^{2} x+\sin^2 x+2 \sin x \cos x\right) d x$

$=\int(1+\sin 2 x) d x$

$=\int 1 d x+\int \sin 2 x d x$

$=x-\frac{\cos 2 x}{2}+c$

Question 39

If $\frac{d y}{d x}=\cos x+\sec ^{2} x$ and where x=0 , y=0 what is y ?

Sol :

$f(x)=\frac{d y}{d x}=\cos x+\sec ^{2} x$

F(x) is antiderivative

∴$F(x)=\int f(x) d x=\int \frac{d y}{d x}=\int \cos x+\sec ^{2} x d x$

F(x)=sinx+tanx

y=sinx+tanx

Question 40

Find the antiderivative F of f defined by f(x)=4x²-6 where F(0)=3

Sol :

Antiderivative F(x)=$\int f(x) d x=\int 4 x^{3}-6 d x$

$F(x)=4 \int x^{3}-6 \int d x$

$=4 \cdot \frac{x^{4}}{4}-6 x+c$

F(x)=x⁴-6x+c

A.T.Q F(0)=0⁴-6.0+c=3

$F(x)=x^{4}-6 x+3$

Question 41

If $f^{\prime}(x)=4 x^{3}-\dfrac{3}{x^4}$ , such that f(2)=0 , then find f(x)

Sol :

$f(x)=\int f^{\prime}(x) d x$

$=\int 4 x^{3}-\frac{3}{x^{4}} d x$

$f(x)=4 \int x^{3} d x-3 \int x^{-4} d x$

$=4 \frac{x^{4}}{4}-3 \cdot \frac{x^{-3}}{-3}+c$

$f(x)=x^{4}+x^{-3}+c$$=x^{4}+\frac{1}{x^{3}}+c$

A.T.Q , f(2)=$(2)^{4}+\dfrac{1}{2^{3}}+c=0$

$\frac{129}{8}+c=0$

$C=-\frac{129}{8}$

$f(x)=x^4+\dfrac{1}{x^3}-\dfrac{129}{8}$