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INDEFINITE INTEGRAL

Total questions: 26

Worksheet time: 4hrs 20mins

Name
Class
Date
1.

 ∫x2−1(x4+3x2+1)Tan−1(x2+1x) \int_{ }^{ }\frac{x^2-1}{\left(x^4+3x^2+1\right)Tan^{-1}\left(\frac{x^2+1}{x}\right)}\   dx is equal to

a)

 Tan−1(x+1x)+cTan^{-1}\left(x+\frac{1}{x}\right)+c  

b)

 log⁡e(Tan−1(x+1x))+c\log_e\left(Tan^{-1}\left(x+\frac{1}{x}\right)\right)+c  

c)

 log⁡e(tan⁡(x2+1x))+c\log_e\left(\tan\left(\frac{x^2+1}{x}\right)\right)+c  

d)

 (x+1x)Tan−1(x+1x)+c\left(x+\frac{1}{x}\right)Tan^{-1}\left(x+\frac{1}{x}\right)+c  

2.

The value of the integral ∫cos⁡3x+cos⁡5xsin⁡2x+sin⁡4xdx is\int_{ }^{ }\frac{\cos^3x+\cos^5x}{\sin^2x+\sin^4x}dx\ is  

a)

 sin⁡x−6Tab−1(sin⁡ x)+c\sin x-6Tab^{-1}\left(\sin\ x\right)+c  

b)

 sin⁡x−2(sin⁡x)−1+c\sin x-2\left(\sin x\right)^{-1}+c  

c)

 sin⁡x−2(sin⁡x)−1−6Tan−1(sin⁡x)+c\sin x-2\left(\sin x\right)^{-1}-6Tan^{-1}\left(\sin x\right)+c  

d)

 sin⁡x−2(sin⁡x)−1+5Tan−1(sin⁡x)+c\sin x-2\left(\sin x\right)^{-1}+5Tan^{-1}\left(\sin x\right)+c  

3.

 ∫x+1x(1+xex)2dx=log⁡e∣xex1+xex∣+f(x)+c,\int_{ }^{ }\frac{x+1}{x\left(1+xe^x\right)^2}dx=\log_e\left|\frac{xe^x}{1+xe^x}\right|+f\left(x\right)+c,  then f(x) is.

a)

 11+xex\frac{1}{1+xe^x}  

b)

 x1+xex\frac{x}{1+xe^x}  

c)

 xex1+x\frac{xe^x}{1+x}  

d)

 xex1+ex\frac{xe^x}{1+e^x}  

4.

 ∫x−1(x+1)x(x2+x+1)dx is\int_{ }^{ }\frac{x-1}{\left(x+1\right)\sqrt{x\left(x^2+x+1\right)}}dx\ is  

a)

 Tan−1(x2+x+1x)+cTan^{-1}\left(\frac{x^2+x+1}{x}\right)+c  

b)

 2Tan−1(x2+x+1x)+c2Tan^{-1}\left(\frac{x^2+x+1}{x}\right)+c  

c)

 Tan−1(x2+x+1x)+cTan^{-1}\left(\frac{\sqrt{x^2+x+1}}{x}\right)+c  

d)

 2Tan−1x+1x+1+c2Tan^{-1}\sqrt{x+\frac{1}{x}+1}+c  

5.

 ∫(x−x3)13x4dx=\int_{ }^{ }\frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4}dx=  

a)

 38(1x2−1)43+c\frac{3}{8}\left(\frac{1}{x^2}-1\right)^{\frac{4}{3}}+c  

b)

 −38(1x2+1)43+c-\frac{3}{8}\left(\frac{1}{x^2}+1\right)^{\frac{4}{3}}+c  

c)

 −38(1x2−1)43+c-\frac{3}{8}\left(\frac{1}{x^2}-1\right)^{\frac{4}{3}}+c  

d)

 −34(1−1x2)43+c-\frac{3}{4}\left(1-\frac{1}{x^2}\right)^{\frac{4}{3}}+c  

6.

 If y(x−y)2=x, then ∫dxx−3yequalsIf\ y\left(x-y\right)^2=x,\ then\ \int_{ }^{ }\frac{dx}{x-3y}equals  

a)

 x2log⁡e{(x−y)2+−1}+c\frac{x}{2}\log_e\left\{\left(x-y\right)^2+-1\right\}+c  

b)

 12log⁡e{(x−y)2−1}+c\frac{1}{2}\log_e\left\{\left(x-y\right)^2-1\right\}+c  

c)

 x+12log⁡e{(x−y)2+1}+cx+\frac{1}{2}\log_e\left\{\left(x-y\right)^2+1\right\}+c  

d)

 log⁡e{(x−y)2−1}+c\log_e\left\{\left(x-y\right)^2-1\right\}+c  

7.

 If ∫(1−x1+x)12 dxx=2 cos⁡−1x−ϕ(x)+cIf\ \int_{ }^{ }\left(\frac{1-\sqrt{x}}{1+\sqrt{x}}\right)^{\frac{1}{2}}\ \frac{dx}{x}=2\ \cos^{-1}\sqrt{x}-\phi\left(x\right)+c  , then  ϕ(x) equals\phi\left(x\right)\ equals  

a)

 log⁡e(1−1−xx)\log_e\left(\frac{1-\sqrt{1-x}}{\sqrt{x}}\right)  

b)

 12log⁡e(1+1−xx)\frac{1}{2}\log_e\left(\frac{1+\sqrt{1-x}}{\sqrt{x}}\right)  

c)

 2log⁡e(1−1−xx)2\log_e\left(\frac{1-\sqrt{1-x}}{\sqrt{x}}\right)  

d)

 2log⁡e(1+1−xx)2\log_e\left(\frac{1+\sqrt{1-x}}{\sqrt{x}}\right)  

8.

 Let I1=∫x1+1(x3+3x+1)13dx=12(x3+3x+1)23+cLet\ I_1=\int_{ }^{ }\frac{x^1+1}{\left(x^3+3x+1\right)^{\frac{1}{3}}}dx=\frac{1}{2}\left(x^3+3x+1\right)^{\frac{2}{3}}+c  and  I2=∫sin⁡2x1+sin⁡2xdx=log⁡e(1+sin⁡2x)+cI_2=\int_{ }^{ }\frac{\sin2x}{1+\sin^2x}dx=\log_e\left(1+\sin^2x\right)+c  Then

a)

 both I1and I2 are correctboth\ I_1and\ I_2\ are\ correct  

b)

 both I1 and I2 are not correctboth\ I_1\ and\ I_2\ are\ not\ correct  

c)

 I1is correct and I2 is notI_1is\ correct\ and\ I_2\ is\ not  

d)

 I2 is correct and I1 is notI_2\ is\ correct\ and\ I_1\ is\ not  

9.

 ∫dx1+x2log⁡e(x+1+x2)is\int_{ }^{ }\frac{dx}{\sqrt{1+x^2}\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)}}is  

a)

 2log⁡e(x+1+x2)+c2\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)+c}  

b)

 log⁡e(x+1+x2)+c\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)+c}  

c)

 12log⁡e(x+1+x2)+c\frac{1}{2}\log_e\left(x+\sqrt{1+x^2}\right)+c  

d)

 12log⁡e(x+1+x2)+c\frac{1}{2}\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)+c}  

10.

 ∫tan⁡x1+tan⁡x+tan⁡2xdx is equal to\int_{ }^{ }\frac{\tan x}{1+\tan x+\tan^2x}dx\ is\ equal\ to  

a)

 x+23Tan−1 (2tan⁡x+1)3+cx+\frac{2}{\sqrt{3}}Tan^{-1}\ \frac{\left(2\tan x+1\right)}{\sqrt{3}}+c  

b)

 x−23Tan−1(2tan⁡x+13)+cx-\frac{2}{\sqrt{3}}Tan^{-1}\left(\frac{2\tan x+1}{\sqrt{3}}\right)+c  

c)

 x+13Tan−1(tan⁡x+13)+cx+\frac{1}{\sqrt{3}}Tan^{-1}\left(\frac{\tan x+1}{\sqrt{3}}\right)+c  

d)

 x−13Tan−1(tan⁡x+13)+cx-\frac{1}{\sqrt{3}}Tan^{-1}\left(\frac{\tan x+1}{\sqrt{3}}\right)+c  

11.

 ∫tan⁡(π4−x)sec⁡2 xtan⁡3x+tan⁡2x+tan⁡xdx equals\int_{ }^{ }\frac{\tan\left(\frac{\pi}{4}-x\right)\sec^2\ x}{\sqrt{\tan^3x+\tan^2x+\tan x}}dx\ equals  

a)

 −2Tan−11+tan⁡x+cot⁡x+c-2Tan^{-1}\sqrt{1+\tan x+\cot x}+c  

b)

 2 Tan−1tan⁡x+cot⁡x+c2\ Tan^{-1}\sqrt{\tan x+\cot x}+c  

c)

 Tan−1tan⁡ x+ cot⁡ x+cTan^{-1}\sqrt{\tan\ x+\ \cot\ x}+c  

d)

 −2Tan−1sec⁡x+tan⁡x+c-2Tan^{-1}\sqrt{\sec x+\tan x}+c  

12.

 ∫2+x(x+x+1)2dx equals\int_{ }^{ }\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx\ equals  

a)

 2x+x+1+c\frac{2}{x+\sqrt{x}+1}+c  

b)

 xx+x+1+c\frac{x}{x+\sqrt{x}+1}+c  

c)

 2x1+x−x+c\frac{2x}{1+\sqrt{x}-x}+c  

d)

 2x1+x+x+c\frac{2x}{1+\sqrt{x}+x}+c  

13.

 ∫2x12+5x9(x5+x3+1)3 dx is equal to\int_{ }^{ }\frac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\ dx\ is\ equal\ to  

a)

 x2+2x(x5+x3+1)2+c\frac{x^2+2x}{\left(x^5+x^3+1\right)^2}+c  

b)

 x102(x5+x3+1)2+c\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+c  

c)

 log⁡e(x5+x3+1)+2x7+5x4+c\log_e\left(x^5+x^3+1\right)+\sqrt{2x^7+5x^4}+c  

d)

 log⁡e(2x7+5x4+x5+x3+1)+c\log_e\left(2x^7+5x^4+\sqrt{x^5+x^3+1}\right)+c  

14.

 If ∫3x4−1(x4+x+1)2dx=∫f1(x)If\ \int_{ }^{ }\frac{3x^4-1}{\left(x^4+x+1\right)^2}dx=\int_{ }^{ }f_1\left(x\right)   dx−∫dxx4+x+1+cdx-\int_{ }^{ }\frac{dx}{x^4+x+1}+c   then  f1(x)isf_1\left(x\right)is  

a)

 4x3+1(x4+x+1)2\frac{4x^3+1}{\left(x^4+x+1\right)^2}  

b)

 x3(4x+1)(x4+x+1)2\frac{x^3\left(4x+1\right)}{\left(x^4+x+1\right)^2}  

c)

 x(4x3−1)(x4+x+1)2\frac{x\left(4x^3-1\right)}{\left(x^4+x+1\right)^2}  

d)

 x(4x3+1)(x4+x+1)2\frac{x\left(4x^3+1\right)}{\left(x^4+x+1\right)^2}  

15.

 ∫(x2+sin⁡2x1+x2) dxcos⁡2xis equal to\int_{ }^{ }\left(\frac{x^2+\sin^2x}{1+x^2}\right)\ \frac{dx}{\cos^2x}is\ equal\ to  

a)

 tan⁡x−Tan−1x+c\tan x-Tan^{-1}x+c  

b)

 tan⁡x+Sin−1x+c\tan x+Sin^{-1}x+c  

c)

 tan⁡ x − Sec−1x+c\tan\ x\ -\ Sec^{-1}x+c  

d)

 cot⁡x−Cot−1x+c\cot x-Cot^{-1}x+c  

16.

 If f : R⟶R is a function such the f(0)=0,f′If\ f\ :\ R\longrightarrow R\ is\ a\ function\ such\ the\ f\left(0\right)=0,f'  (0) = 3 and it satisfies the relation  f(x+y3)=f(x)+f(y)3f\left(\frac{x+y}{3}\right)=\frac{f\left(x\right)+f\left(y\right)}{3}   for all x, y  x, y ∈R then ∫f(x)dx equalsx,\ y\ \in R\ then\ \int_{ }^{ }f\left(x\right)dx\ equals  

a)

 3x2+33x^2+3  

b)

 2x23+3\frac{2x^2}{3}+3  

c)

 32x2+c\frac{3}{2}x^2+c  

d)

 3x2+c3x^2+c  

17.

 ∫x4+1x6+1dx is equal to\int_{ }^{ }\frac{x^4+1}{x^6+1}dx\ is\ equal\ to  

a)

 Tan−1x−13Tan−1(x3)+cTan^{-1}x-\frac{1}{3}Tan^{-1}\left(x^3\right)+c  

b)

 Tan−1x+Tan−1(x3)3+Tan^{-1}x+\frac{Tan^{-1}\left(x^3\right)}{3}+  

c)

 Tan−1x+Tan−1(x2)2+cTan^{-1}x+\frac{Tan^{-1}\left(x^2\right)}{2}+c  

d)

 Tan−1x−Tan−1(x2)+c2Tan^{-1}x-\frac{Tan^{-1}\left(x^2\right)+c}{2}  

18.

 ∫sin⁡3xdx(1+cos⁡2x)1+cos⁡2x+cos⁡4xequals\int_{ }^{ }\frac{\sin^3xdx}{\left(1+\cos^2x\right)\sqrt{1+\cos^2x+\cos^4x}}equals  

a)

 Cos−1(sec⁡ x + cos⁡ x) +cCos^{-1}\left(\sec\ x\ +\ \cos\ x\right)\ +c  

b)

 Sin−1(sec⁡ x + cos⁡ x)+cSin^{-1}\left(\sec\ x\ +\ \cos\ x\right)+c  

c)

 Sec−1(sec⁡ x + cos⁡ x)+cSec^{-1}\left(\sec\ x\ +\ \cos\ x\right)+c  

d)

 Tan−1(sec⁡ x + tan⁡ x)+cTan^{-1}\left(\sec\ x\ +\ \tan\ x\right)+c  

19.

 (1+x)sin⁡x(x2+2x)cos⁡2x−(1+x)sin⁡ 2xdx=12log⁡e\frac{\left(1+x\right)\sin x}{\left(x^2+2x\right)\cos^2x-\left(1+x\right)\sin\ 2x}dx=\frac{1}{2}\log_e   ∣t+1t−1∣+c where t is\left|\frac{t+1}{t-1}\right|+c\ where\ t\ is  

a)

(x + 1 cos x - sin x

b)

(x + 1) sin x - cos x

c)

(x + 1) sin x + cos x

d)

(x + 1) cos x + sin x

20.

If f is a real-valued function satisfying the relation 5f(x)+3f(1x)=x+2 for all x≠0, then5f\left(x\right)+3f\left(\frac{1}{x}\right)=x+2\ for\ all\ x\ne0,\ then  ∫xf (x) dx is.\int_{ }^{ }xf\ \left(x\right)\ dx\ is.   


a)

 16(53x3+2x2+3x)+c\frac{1}{6}\left(\frac{5}{3}x^3+2x^2+3x\right)+c  

b)

 116(53x3−2x2+3x)+c\frac{1}{16}\left(\frac{5}{3}x^3-2x^2+3x\right)+c  

c)

 116(5x33+2x2−3x)+c\frac{1}{16}\left(\frac{5x^3}{3}+2x^2-3x\right)+c  

d)

 116(53x3−2x2−3x)+c\frac{1}{16}\left(\frac{5}{3}x^3-2x^2-3x\right)+c  

21.

 ∫2x5+x4−2x3+2x2+1(x2+1)(x4−1)dx(x>1)is equal to\int\frac{2x^5+x^4-2x^3+2x^2+1}{\left(x^2+1\right)\left(x^4-1\right)}dx\left(x>1\right)is\ equal\ to  

a)

 log⁡e(x2+1)+12log⁡e(x−1x+1)+c\log_e\left(x^2+1\right)+\frac{1}{2}\log_e\left(\frac{x-1}{x+1}\right)+c  

b)

 log⁡e(x2+1x2−1)+log⁡e(x−1x+1)+c\log_e\left(\frac{x^2+1}{x^2-1}\right)+\log_e\left(\frac{x-1}{x+1}\right)+c  

c)

 log⁡e(x2+1x2−1)+12log⁡e(x−1x+1)+c\log_e\left(\frac{x^2+1}{x^2-1}\right)+\frac{1}{2}\log_e\left(\frac{x-1}{x+1}\right)+c  

d)

 log⁡e(x2+1)+1x2+1+12log⁡e(x−1x+1)+c\log_e\left(x^2+1\right)+\frac{1}{x^2+1}+\frac{1}{2}\log_e\left(\frac{x-1}{x+1}\right)+c  

22.

 If f(x)=xand g(x)=ex−1, then If\ f\left(x\right)=\sqrt{x}and\ g\left(x\right)=e^x-1,\ then\    ∫(f o g) (x) dx equals\int_{ }\left(f\ o\ g\right)\ \left(x\right)\ dx\ equals  

a)

 2ex−1−2 Tan−1(ex−1)+c2\sqrt{e^x-1}-2\ Tan^{-1}\left(\sqrt{e^x-1}\right)+c  

b)

 2(ex−1)−2 Tan−1(ex−1)+c2\left(e^x-1\right)-2\ Tan^{-1}\left(\sqrt{e^x-1}\right)+c  

c)

 2ex−1−2 tan⁡(ex−1)+c2\sqrt{e^x-1}-2\ \tan\left(\sqrt{e^x-1}\right)+c  

d)

 2ex−1−2 Tan −1(ex−1)+c2\sqrt{e^x-1}-2\ Tan\ ^{-1}\left(e^x-1\right)+c  

23.

 ∫dxx2(x4+1)34equals\int_{ }^{ }\frac{dx}{x^2\left(x^4+1\right)^{\frac{3}{4}}}equals  

a)

 (x4+1)14+c\left(x^4+1\right)^{\frac{1}{4}}+c  

b)

 −(x4+1)14x4+c-\frac{\left(x^4+1\right)^{\frac{1}{4}}}{x^4}+c  

c)

 −(x4+1)34x3+c-\frac{\left(x^4+1\right)^{\frac{3}{4}}}{x^3}+c  

d)

 (x3+1)34x4+c\frac{\left(x^3+1\right)^{\frac{3}{4}}}{x^4}+c  

24.

 If ∫x3+3x+2(x2+1)2(x+1)dx =−12log⁡eIf\ \int_{ }^{ }\frac{x^3+3x+2}{\left(x^2+1\right)^2\left(x+1\right)}dx\ =-\frac{1}{2}\log_e    ∣x+1∣+14log⁡e(x2+1)+f(x)+c then f(x) equals\left|x+1\right|+\frac{1}{4}\log_e\left(x^2+1\right)+f\left(x\right)+c\ then\ f\left(x\right)\ equals  

a)

 12 Tan−1x+x1+x2\frac{1}{2}\ Tan^{-1}x+\frac{x}{1+x^2}  

b)

 32Tan−1x−x1+x2\frac{3}{2}Tan^{-1}x-\frac{x}{1+x^2}  

c)

 32Tan−1x+x1+x2\frac{3}{2}Tan^{-1}x+\frac{x}{1+x^2}  

d)

 32log⁡e∣x∣+x1+x2\frac{3}{2}\log_e\left|x\right|+\frac{x}{1+x^2}  

25.

 If ∫1−(cot⁡x)2010tan⁡x+(cot⁡x)2011dx=1klog⁡eIf\ \int_{ }^{ }\frac{1-\left(\cot x\right)^{2010}}{\tan x+\left(\cot x\right)^{2011}}dx=\frac{1}{k}\log_e    ∣(sin⁡x)k+(cos⁡x)k∣+c then k is equal to\left|\left(\sin x\right)^k+\left(\cos x\right)^k\right|+c\ then\ k\ is\ equal\ to  

a)

2010

b)

2011

c)

2012

d)

2013

26.

 ∫1+xsec⁡2xx(xetan⁡x+1)dx is equal to\int_{ }^{ }\frac{1+x\sec^2x}{x\left(xe^{\tan x}+1\right)}dx\ is\ equal\ to  

a)

 log⁡e∣xetan⁡xxetan⁡x+1∣+c\log_e\left|\frac{xe^{\tan x}}{xe^{\tan x}+1}\right|+c  

b)

 log⁡e∣(x+1)etan⁡xextan⁡ x+1∣+c\log_e\left|\frac{\left(x+1\right)e^{\tan x}}{e^{x\tan\ x}+1}\right|+c  

c)

 log⁡e∣extan⁡x+1xetan⁡x+1∣+c\log_e\left|\frac{e^{x\tan x}+1}{xe^{\tan x}+1}\right|+c  

d)

 log⁡e∣xetan⁡x+1(x+1)etan⁡x+1∣+c\log_e\left|\frac{xe^{\tan x}+1}{\left(x+1\right)e^{\tan x}+1}\right|+c  

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