Mathematics previous year question of Indefinite Integral for NDA

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previous year question of Indefinite Integral for NDA

Previous Year Indefinite Integral Questions For NDA

Set - 1

Q 2783491347

What is `int (dx)/( x(x^7+1))` equal to?
NDA Paper 1 2017

(A)

`1/2 ln | (x^7-1)/(x^7+1)|+c`

(B)

`1/7 ln | (x^7+1)/(7x)|+c`

(C)

`1/7 ln | (x^7-1)/(7x)|+c`

(D)

`1/7 ln | (x^7)/(x^7+1)|+c`

Solution:

`I= int(dx)/(x(x^7+1)) =int(dx)/(x^8 (1+1/x^7))`

`t= 1+ 1/x^7`

`dt =(-7 dx)/(x^8)`

`I= 1/(-7) int (1/t) dt`

`=1/(-7) lnt +C`

`=1/(-7) ln |(x^7 +1)/(x^7)| +C`

Correct Answer is `=>` (D) `1/7 ln | (x^7)/(x^7+1)|+c`

Q 2773591446

What is `int (( x^(e-1) + e^(x-1) )dx)/(x^e +e^x)`
NDA Paper 1 2017

(A)

`x^2/2+c`

(B)

`ln (x+e) +c`

(C)

`ln (x^e+e^x)+c`

(D)

`1/e ln (x^e+e^x) +c`

Solution:

Let `I= int(x^(e+1) +e^(x-1))/(x^e +e^x) dx`

Let `t= x^e + e^x`

`dt = (ex^(e-1) + e^x ) dx`

`(dt)/e =(x^(e-1) + e^(x-1)) dx`

`=> I= int 1/(te) dt => 1/e ln (x^e+ e^x)`

Correct Answer is `=>` (D) `1/e ln (x^e+e^x) +c`

Q 2761267125

What is `int e^(sinx) ( x cos^3 x - sin x)/( cos^2 x) dx` equal to ?
NDA Paper 1 2016

(A)

`( x+secx) e^(sinx)+C`

(B)

`(x- secx) e^(sinx)+C`

(C)

`(x+tanx)e^(sinx)+C`

(D)

`(x-tanx)e^(sinx)+C`

Solution:

`int e^(sinx) [ x cos x - -tan x sec x] dx`

`int e^(sin x) [ x cos x -1] + int e^(sin x) [1- sec x tan x]`

`int (e^(sin x) cos x [x- sec x] + e^(sin x) [ 1- sec x tan x]) dx`

`int [( x e^(sin x) - e^(sin x) ) +(e^(sin x) cos x * sec x- e^(sin x) sec x tan x)]+ C`

`=int d/(dx) ( x e^(sin x))- int d/(dx) ( sec x e^(sin x))+ C`

`= (x- sec x)e^(sin x) + C`

Correct Answer is `=>` (B) `(x- secx) e^(sinx)+C`

Q 1668391205

` int (dx)/( a cos x + b sin x)` is of the form `1/r ln [ tan ( (x+a)/2) ]`

What is `r` equal to?
NDA Paper 1 2015

(A)

`a^ 2 + b^2`

(B)

`sqrt(a^ 2 + b^2)`

(C)

`a + b`

(D)

`sqrt(a^ 2 - b^2)`

Solution:

Let ` int (dx)/( a cos x + b sin x)`

put `a = r sin alpha`

and `b = r cos alpha`

where `r = sqrt(a^ 2 + b^2)`

Correct Answer is `=>` (B) `sqrt(a^ 2 + b^2)`

Q 1628591401

` int (dx)/( a cos x + b sin x)` is of the form `1/r In [ tan ( (x+a)/2) ]`

What is `alpha` equal to?
NDA Paper 1 2015

(A)

` tan^(-1) (a/b)`

(B)

` tan^(-1) (b/a)`

(C)

` tan^(-1) ((a+b)/(a-b))`

(D)

` tan^(-1) ((a-b)/(a+b))`

Solution:

Let ` int (dx)/( a cos x + b sin x)`

put `a = r sin alpha`

and `b = r cos alpha`

and ` alpha = tan^(-1) (a/b)`

`:. I = 1/r int (dx)/( sin alpha cos x + cos alpha sin x)`

` = 1/r int (dx)/( sin(x +alpha) ) = 1/r int cosec (x +alpha) dx`

`=1/r In (cosec (x + alpha)- cot (x +alpha))+ C`

` = 1/r In ( 1/ ( sin(x +alpha) ) - ( cos (x + a))/( sin(x +alpha) )) + C`

` = 1/r In ((1- cos (x + alpha ))/( sin(x +alpha) )) + C`

`= 1/r In ( ( 2 sin^2 ( x+alpha)/2)/ ( (2 sin)( x+alpha)/2 (cos) ( x+alpha)/2)) + C`

`=1/r In ((tan) ( x+alpha)/2) + c`

Correct Answer is `=>` (A) ` tan^(-1) (a/b)`

Q 1639412312

What is `int (dx)/sqrt( x^2 + a^2)` equal to?

NDA Paper 1 2015

(A)

` ln | ( x + sqrt( x^2 + a^2))/a | + C`

(B)

` ln | ( x - sqrt( x^2 + a^2))/a | + C`

(C)

` ln | ( x^2 + sqrt( x^2 + a^2))/a | + C`

(D)

None of these

Solution:

Let `I = int (dx)/sqrt( x^2 + a^2)`

`= ln |x + sqrt( x^2 + a^2) | + C`,

where `C` is the constant of integration.

Correct Answer is `=>` (C) ` ln | ( x^2 + sqrt( x^2 + a^2))/a | + C`

Q 2231380222

` int (dx)/(1 + e^(-x))` is equal to

where, `C` is the constant of integration.
NDA Paper 1 2015

(A)

`1 +e^x + C`

(B)

`ln (1 + e^(-x)) + C`

(C)

`ln(1 + e^(x)) + C`

(D)

` 2ln (1 + e^(-x)) + C`

Solution:

Let `I = int (dx)/(1 + e^(-x)) = int (dx)/(1 + 1/e^(x)) = int (e^x dx)/(1 + e^(x))`

Put `t = 1 + e^x`

`=> dt = e^x dx`

Now ` I = int (dt)/t = ln (t) _C =ln (1+e^x) + C quad [∵ t = 1 + e^x]`

Correct Answer is `=>` (C) `ln(1 + e^(x)) + C`

Q 1649134913

What is `int ( xe^x dx)/(x+1)^2 ` equal to?

where, `C` is the constant of integration.
NDA Paper 1 2015

(A)

`(x + 1)^2 e^x + C`

(B)

`(x + 1) e^x + C`

(C)

` e^x/(x+1) + C`

(D)

` e^x/(x+1)^2 + C`

Solution:

Let `I = int (xe^x)/(x+1)^2 dx = int e^x ( ((x+1) - 1)/ (x+1)^2) dx`

` = int e^x ( 1/ (x+1) + ( (-1)/ (x+1)^2))dx`

` => e^x ( 1/( x + 1) ) + C quad ( :. int e^x (f(x) + f'(x))dx = e^x f(x) +C)`

Correct Answer is `=>` (C) ` e^x/(x+1) + C`

Q 1771701626

Consider the function `f' '(x) = sec^4 x + 4` with `f(0) = 0` and
`f'(0) =0`.

What is `f' (x)` equal to?
NDA Paper 1 2014

(A)

`tan x - (tan^3 x)/3 + 4x`

(B)

`tan x + (tan^3 x)/3 + 4x`

(C)

`tan x + (sec^3 x)/3 + 4x`

(D)

`- tan x - (tan^3 x)/3 + 4x`

Solution:

Clearly, `f ' (x) = int f' '(x) dx + C_1`

`= int (sec^4 x+ 4) dx + C_1`

` = int sec^2 x sec^2 x dx + int 4 dx + C_1`

`= int (1 + tan^2 x) sec^2 xdx + 4x + C_1`

` = I_1 + 4x + C_1`

Put `tan x = t` in the integra `I_1`, then

`sec^2 x dx = dt`

`:. I_1 =int (1+t^ 2) dt = t + (t^3)/3 + C'`

`= tan x + ( tan^3 x)/3 + C'`

`:. f'(x) = tan x + ( tan^3 x)/3 + 4x + C`

Where `C = C_1 + C'`

`∵ f' (x) = 0 => C = 0`

Thus `f'(x) = tan x + ( tan^3 x)/3 + 4x`

Correct Answer is `=>` (B) `tan x + (tan^3 x)/3 + 4x`

Q 1711312220

Consider
`int x tan^(-1) x dx = A(x^2 + 1) tan^(-1) x + Bx + C`
where, `C` is the constant of integration.

What is the value of `A?`
NDA Paper 1 2014

(A)

`1`

(B)

`1/2`

(C)

`-1/2`

(D)

`1/4`

Solution:

Given, `int x tan^(-1) x dx = A(x^2 + 1)tan^(-1) x + Bx + C`

where, `C` is the constant of Integration

Consider, ` int x tan^(-1) x dx`

` = tan^(-1) x . x^2/2 - int d/(dx) (tan^(-1) x) . x^2/2 dx`

(using integration by parts)

` = (x^2 .tan^(-1) x)/2 - 1/2 int x^2/(1 + x^2) dx`

` = (x^2 .tan^(-1) x)/2 - 1/2 ( int ( (1 + x^2 - 1)/( 1 + x^2))dx )`

` = (x^2 .tan^(-1) x)/2 - 1/2 ( int dx - int (dx)/( 1 + x^2))`

` = (x^2 .tan^(-1) x)/2 - 1/2 (x - tan^(-1) x) + C`

` = (x^2 .tan^(-1) x)/2 - x/2 + (tan^(-1))/2 + C`

` = 1/2 (x^2 + 1) tan^(-1) x - x/2 + C`

Clearly, `A = 2`, hence option `(b)` is correct.

Correct Answer is `=>` (B) `1/2`

Q 1751312224

Consider
`int x tan^(-1) x dx = A(x^2 + 1) tan^(-1) x + Bx + C`
where, `C` is the constant of integration.

What is the value of `B?`
NDA Paper 1 2014

(A)

`1`

(B)

`1/2`

(C)

`- 1/2`

(D)

`1/4`

Correct Answer is `=>` (C) `- 1/2`

Q 2329523411

What is `int (dx)/(sqrt(4+x^2))` equal to?

where, `C` is an arbitrary constant
NDA Paper 1 2013

(A)

`log | sqrt(4+x^2)+x|+C`

(B)

`log | sqrt(4+x^2)-x|+C`

(C)

`sin^(-1)(x/2)+C`

(D)

None of the above

Solution:

Let `I=int (dx)/(sqrt(4+x^2))=int (dx)/(sqrt(x^2+(2)^2)`

`=log | {x+sqrt(x^2+4)}|+C`

Correct Answer is `=>` (A) `log | sqrt(4+x^2)+x|+C`

Q 2379523416

What is `int sin^2 xdx + int cos^2 xdx` equal to?

NDA Paper 1 2013

(A)

`x+C`

(B)

`x^2/2 +C`

(C)

`x^2+C`

(D)

None of these

Solution:

Let `I=int sin^2 x dx + int cos^2 x dx`

`=int (sin^2 x+ cos^x)dx`

`=int 1 * dx=x+C` `(sin^2 theta +cos^2 theta=1)`

Correct Answer is `=>` (A) `x+C`

Q 2339623512

What is `int e^(e^x) e^x dx` equal to?
NDA Paper 1 2013

(A)

`e^(e^x)+C`

(B)

`2e^(e^x)+C`

(C)

`e^(e^x) e^x+C`

(D)

`2e^(e^x) e^x+C`

Solution:

Let `I=int e^(e^x) * e^x dx` (put `t=e^x=> dt=e^x dx`)

`= int e^t * dt =e^t+C=e^(e^x)+C`

Correct Answer is `=>` (A) `e^(e^x)+C`

Q 2329723611

What is `int (x cos x+ sin x) dx` equal to?
NDA Paper 1 2013

(A)

`x sin x+C`

(B)

`x cos x+C`

(C)

`-x sin x+C`

(D)

`-x cos x+C`

Solution:

Let `I=int (x cos x+sin x) dx=int cos x dx+ int sin x dx`

`=x sin x- int sin x dx+ int sin x dx`

`= x sin x +C`

Correct Answer is `=>` (A) `x sin x+C`

Q 2379723616

What is `int (dx)/(x ln x)` equal to?

NDA Paper 1 2013

(A)

`ln (ln x)+C`

(B)

`ln x +C`

(C)

`(ln x)^2+C`

(D)

None of these

Solution:

Let `I=int (dx)/(x log x)`

Let `{ tt((t=logx),(dt=1/x dx))=> I=int 1/t * dt=log|t|+C`

`:. I=log (log x)+C`

Correct Answer is `=>` (A) `ln (ln x)+C`

Q 2379823716

What is `int e^(ln x) dx` equal to?
NDA Paper 1 2013

(A)

`x e^(|ln x|)+C`

(B)

`-x e^(|-ln x|)+C`

(C)

`x+C`

(D)

`x^2/2+C`

Solution:

Let `I=int e^(log x) dx` (by logarithm proptrty, `e^(loga) =a`)

`:. I=int x dx=[x^2/2]+C`

Correct Answer is `=>` (D) `x^2/2+C`

Q 2329023811

What is the value of `int a^x e^x dx ` ?
NDA Paper 1 2012

(A)

`(a^x e^x)/(ln a)+C`

(B)

`a^x e^x+C`

(C)

`(a^xe^x)/(ln(ae))+C`

(D)

None of these

Solution:

Let `I=int a^x * e^x dx =a^x int e^x dx- int [(d/(dx) a^x) int e^x dx] dx`

`=a^x * e^x-int a^x * log a * e^x dx +C`

`=a^x*e^x-log a int a^x * e^x dx =a^x * e^x +C`

`=> int a^x * e^x dx = (a^x e^x)/(log_e e+log_e a)+C`

`:. I=(a^x e^x)/(log (ae))+C`

Correct Answer is `=>` (C) `(a^xe^x)/(ln(ae))+C`

Q 2319023819

What is the value of `int (ln x)/x dx` ?
NDA Paper 1 2012

(A)

`(ln x)^2/2 +C`

(B)

`((ln x))/2+C`

(C)

`(ln x)^2+C`

(D)

None of these

Solution:

Let `I=int (ln x)/x dx=> I= int t *dt`

`:. I=t^2/2+C=((ln x)^2)/2+C`

Correct Answer is `=>` (A) `(ln x)^2/2 +C`

Q 2389123917

What is the value of `int (1/(cos^2x)-1/(sin^2x)) dx` ?
NDA Paper 1 2012

(A)

`2 cosec 2x + C`

(B)

`- 2 cot 2x + C`

(C)

`2 sec 2x + C`

(D)

`- 2 tan 2x + C`

Solution:

`int (1/(cos^2x)-1/(sin^2x)) dx=int (sec^2x+cosec^2x)dx`

`=tan x-(-cot x)+C=tan x+cot x+C`

`=((sinx)/(cos x)+(cos x)/(sin x))+C=((sin^2x+ cos^2 x)/(sin x * cos x))+C`

`=(2/(sin x * cos x))+C=(2/(sin2 x))+C`

`=2 cosec 2x+C`

Correct Answer is `=>` (A) `2 cosec 2x + C`

Q 2379134016

What is the value of `int (x^2+1)^(5//2) x dx`?
NDA Paper 1 2012

(A)

`(x^2+1)^(7//2)+C`

(B)

`2/7 (x^2+1)^(7//2)+C`

(C)

`1/7 (x^2+1)^(7//2)+C`

(D)

None of these

Solution:

Let `I=int (x^2+1)^(5//2) x dx` (Let `t=x^2+1)=> dt=2xdx`)

`:. I=int t^(5//2) *(dt)/2 =1/2 [(t^(7//2))/(7//2)]+C=1/7(x^2+1)^(7//2)+C`

Correct Answer is `=>` (C) `1/7 (x^2+1)^(7//2)+C`

Q 2369234115

Consider the following statements

I. `int log 10 dx= x + C`

II. `int 10^x dx=10x+C`

Which of the above statements is/are correct?
NDA Paper 1 2012

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

I. `int 10 dx =int 1 * dx =x+C`

II. `int 10^x dx =(10^x)/(log_e 10)+C`

Correct Answer is `=>` (A) Only I

Q 2359534414

What is `int (dx)/(sin^2x cos^2 x)` equal to ?
NDA Paper 1 2011

(A)

`tan x+cot x+C`

(B)

`tan x-cot x+C`

(C)

`(tan x+cot x)^2+C`

(D)

`(tan x-cot x)^2+C`

Solution:

Given, `int (dx)/(sin^2 x * cos^2x)=4 int (dx)/((2 sin * cos x)^2)`

`=4 int (dx)/((sin^2x)^2) =4 int cosec^2 2x dx`

`=- 4 (cot 2x)/2 +c=-2 cot 2x+C`

`=-(2 cos 2x)/(sin 2x)=(-2(cos^2x-sin^2x))/(2 sin x * cos x)+C`

`=-(cot x -tan x)+C=(tan x-cot x)+C`

Correct Answer is `=>` (B) `tan x-cot x+C`

Q 2339634512

What is `int e^x(sqrt x+1/(2 sqrt x)) dx` equal to?
NDA Paper 1 2011

(A)

`xe^x+C`

(B)

`e^x(sqrt x)+C`

(C)

`2e^x(sqrt x)+C`

(D)

`2xe^x+C`

Solution:

`int e^x(sqrt x+1/(2 sqrt x)) dx`

`=int e^x * sqrt x dx+int e^x * 1/(2 sqrt x) dx`

`=e^x * sqrt x-int e^x * 1/(2 sqrt x) dx + int e^x 1/(2 sqrt x)=e^x * sqrt x +C`

Correct Answer is `=>` (B) `e^x(sqrt x)+C`

Q 2319634519

What is `int (sin sqrt x)/(sqrt x)dx` equal to?
NDA Paper 1 2011

(A)

`(cos sqrt x)/2 +C`

(B)

`2 cos sqrt x+C`

(C)

`-(cos sqrt x)/2 +C`

(D)

`-2 cos sqrt x+C`

Solution:

Let `I=int (sin sqrt x)/(sqrt x) dx`

Put `sqrt x=I=> 1/(2 sqrt x) dx=dt => 1/(sqrt x) dx =2 dt`

`:. I=2 int sin t dt =-2 cos t +C=-2 cos sqrt x+C`

Correct Answer is `=>` (D) `-2 cos sqrt x+C`

Q 2359734614

What is `int sin^(-1)(cos x) dx` equal to?
NDA Paper 1 2011

(A)

`(x pi)/2-x^2/2+C`

(B)

`pi/2+x^2/2+C`

(C)

`-(x pi)/2-x^2/2+C`

(D)

`pi/2-x^2/2+C`

Solution:

Let `I=int sin^(-1)(cos x) dx`

`=int sin^(-1)[sin ( pi/2-x)]dx=int(pi/2-x) dx=(pi x)/2-x^2/2+C`

Correct Answer is `=>` (A) `(x pi)/2-x^2/2+C`

Q 2369034815

What is `int sqrt x e^(sqrt x) dx` equal to?
NDA Paper 1 2010

(A)

`2e^(sqrt x) (x -2sqrt x + 2)+ C`

(B)

`2e^(sqrt x) (x + 2sqrt x+ 2) + C`

(C)

`2e^(sqrt x) (x + 2 sqrt x -2)+ C`

(D)

`2e ^(sqrt x) (x- 2sqrt x- 2) + C`

Solution:

Let `I=int sqrt x e^(sqrt x) dx`

Put `sqrt x=t=> 1/(2 sqrt x) dx =dt => dx =2t dt`

`:. I=int t e^t 2t dt=2 int t^2 e^t dt =2 (t^2 e^t-int 2t e^t dt)`

`= 2 [t^ 2e^t- 2(te^t-int e^t* dt)]`

`= 2 (t^2 e^t -2te^t + 2e^t) + C`

`=2(xe^(sqrt x) -2sqrt x e^(sqrt x)+ 2e^(sqrt x))+ C`

`=2e^(sqrt x) (x- 2sqrt x + 2)+ C`

Correct Answer is `=>` (A) `2e^(sqrt x) (x -2sqrt x + 2)+ C`

Q 2359134914

What is `int sec^n x tan x dx` equal to?
NDA Paper 1 2010

(A)

`(sec^n x)/n+C`

(B)

`(sec^(n-1))/(n-1)+C`

(C)

`(tan^n x)/n+C`

(D)

`(tan^(n-1) x)/(n-1)+C`

Solution:

Let `I= int sec^n x tan x dx`

Put `sec x = t =>sec x tan x dx = dt`

`:. I = int t^( n-1) dt = t^n/n + C = (sec^n x)/n + C`

Correct Answer is `=>` (A) `(sec^n x)/n+C`

Q 2379145016

What is `int (e^x(1+x))/(cos^2(x e^x))dx` equal to?
NDA Paper 1 2010

(A)

`xe^x+C`

(B)

`cos (xe)^x+C`

(C)

`tan (xe^x)+C`

(D)

`x cosec(xe^x)+C`

Solution:

Let `I=int (e^x(1+x))/(cos^2(x e^x)) dx`

Put `xe^x=t=> e^x (1+x)dx=dt`

`:. I=int sec^2t dt=tan t+C=tan (xe^x)+C`

Correct Answer is `=>` (C) `tan (xe^x)+C`

Q 2319145019

What is `int e^(ln x) sinx dx ` equal to?
NDA Paper 1 2010

(A)

`e^(ln x) (sin x-cos x)+C`

(B)

`(sin x-x cos x)+C`

(C)

`(x sin x+cos x)+C`

(D)

`(sin x+x cos x)-C`

Solution:

`int e^(lnx) sin x dx=int x sin x dx` (`e^(log a)=a`)

`=- x cos x + int 1 * cos xdx = sin x - x cos x + C`

Correct Answer is `=>` (B) `(sin x-x cos x)+C`

Q 2319245119

What is `int (x^4+1)/(x^2+1) dx` equal to?
NDA Paper 1 2010

(A)

`x^3/3-x+4 tan^(-1) x+C`

(B)

`x^3/3+x+4 tan^(-1)+C`

(C)

`x^3/3-x+2 tan^(-1) x+C`

(D)

`x^3/3-x-4 tan^(-1) x+C`

Solution:

`int (x^4+1)/(x^2+1) dx =int ((x^4-1)/(x^2+1)+2/(x^2+1)) dx`

`int (x^2-1+2/(x^2+1))dx =x^3/3-x+2 tan^(-1) x+C`

Correct Answer is `=>` (C) `x^3/3-x+2 tan^(-1) x+C`

Q 2319445310

If `int x^2 ln x dx = x^3/m ln x +x^3/n + C`, then
the values of m and n, respectively?
NDA Paper 1 2010

(A)

`1/3` and `-1/9`

(B)

`3` and `-9`

(C)

`3` and `9`

(D)

`3` and `3`

Solution:

`int x^2 ln x dx=ln x x^3/3-int 1/x * x^3/3 dx`

`=x^3/3 ln x -int x^2/3 dx =x^3/3 ln x-1/3 * x^3/3 +C`

`=x^3/3 ln x-x^3/9+C`

But `int x^2ln x dx=x^3/m ln x+x^3/n +C` (on comparing)

`:. m=3` and `n=-9`

Correct Answer is `=>` (B) `3` and `-9`

Q 2359445314

What is `int 1/(1+e^x) dx` equal to?
NDA Paper 1 2010

(A)

`x-log x +C`

(B)

`x-log (tan x)+C`

(C)

`x-log(1+e^x)+C`

(D)

`log (1+e^x)+C`

Solution:

`int 1/(1+e^x) dx=int (e^(-x)/(e^(-x)+1) )dx`

`=-log(1+e^(-x)+C=-log ((1+e^x)/e^x)+C`

`=-{log(1+e^x)-log e^x}+C=x-log(1+e^x)+C`

Correct Answer is `=>` (C) `x-log(1+e^x)+C`

Q 2309445318

What is `int(a+b sin x)/(cos^2x)dx` equal to?
NDA Paper 1 2009

(A)

`a sec x + b tan x + C`

(B)

`a tan x + b sec x+C`

(C)

`a cot x + b cosec x + C`

(D)

`a cosec x + b cot x+C`

Solution:

`int (a+b sin x)/(cos^2 x) dx =int(a sec^2 x+b tan x sec x) dx`

`=a tan x+b sec x+C`

Correct Answer is `=>` (B) `a tan x + b sec x+C`

Q 2319545419

What is `int (log x)/((1+log x)^2) dx` equal to
NDA Paper 1 2009

(A)

`1/((1+log x)^3)+C`

(B)

`1/((1+log x)^2)+C`

(C)

`x/((1+log x))+C`

(D)

`x/((1+log x)^2)+C`

Solution:

Let `I=int (log x)/((1+log x)^2)dx`

Put `log x=t=> 1/x dx=dt` and `x=e^t`

`:. I=int (e^t *t)/((1+t)) dt-int(e^t)/((1+t)^2)dt`

`=e^t/(1+t)-int e^t 1/((1+t)^2) dt - int e^t/((1+t)^2) dt =x/(1+log x)+C`

Correct Answer is `=>` (C) `x/((1+log x))+C`

Q 2369745615

NDA Paper 1 2009

Assertion : `int e^x/x(1+x log x)dx=e^x log x+C`

Reason : `int e^x[f(x)+f'(x)]dx=e^xf(x)+C`

(A)

Both A and R individually true and R is the correct explanation of A

(B)

Both A and R are individually true but R is not the correct explanation of A

(C)

A is true but R is false

(D)

A is false but R is true

Solution:

A. `int e^x/x (1+x log x)dx=int e^x/x dx+int log x dx`

`=e^x log x - int e^x log x dx + int e^x log x dx`

`=e^x log x+C`

R. `int e^x[f(x)+f'(x)] dx=intf(x)dx+int e^x f'(x)dx`

`=e^x f(x)-int e^x f'(x) dx+int e^x f'(x)dx`

`=e^xf(x)+C`

Hence both A and R are true is the correct explanation of A

Correct Answer is `=>` (A)

Q 2379845716

What is `int tan^2 x sec^4 x dx` equal to?
NDA Paper 1 2009

(A)

`(sec^5 x)/5+(sec^3 x)/3+C`

(B)

`(tan^5x)/5+(tan^3 x)/3+C`

(C)

`(tan^5 x)/5+(sec^3 x)/3+C`

(D)

`(tan^5 x)/5+(sec^3 x)/3-C`

Solution:

Let `I=int tan^2 x sec ^4 x dx`

`=int tan^2 x(1+tan^2 x)sec^x dx`

Let `tan x=t` and `sec^2 x dt=int (t^2+t^4)dt`

`=t^5/5+t^3/3+C=(tan^5 x)/5+(tan^3 x)/3+C`

Correct Answer is `=>` (B) `(tan^5x)/5+(tan^3 x)/3+C`

Q 2329045811

What is `int sec x ^o dx` equal to?
NDA Paper 1 2009

(A)

`log (sec x^o +tan x ^o ) + C`

(B)

`(180^o log tan (pi/4+(pi x)/(360^o)))/(180^o) +C`

(C)

`(180^o log tan (pi/4+( x)/(2)))/pi +C`

(D)

`(180^o log tan (pi/4+(pi x)/(360^o)))/pi +C`

Solution:

Let `I=int sec x^o dx`

`=int sec(pi x)/(180^o) dx`

Put `(pi x)/(180^o)=t=>dx=(180^o)/pi dt`

`:. I=int sec t dt * (180^o)/pi`

`=(180^o)/pi log tan (pi/4+t/2)+C`

`=(180^o)/pi log (pi/4+(pi x)/(360^o))+C`

Correct Answer is `=>` (D) `(180^o log tan (pi/4+(pi x)/(360^o)))/pi +C`

Q 2339156012

What is `int (e^x+1)^(-1) dx` equal to?
NDA Paper 1 2008

(A)

`ln (ex + 1) + C`

(B)

`ln (e^(-x) + 1) + C`

(C)

`-ln (e^(-x) + 1) + C`

(D)

`-(ex + 1) + C`

Solution:

Let `I=int(e^x+1)^(-1) dx=int1/(e^x+1) dx =int (e^(-x))/(1+e^(-x)) dx`

Let `1+e^(-x)=t=> e^(-x) dx=dt`

`:. I=-int 1/t dt =-log t+C=-log (1+e^(-x))+C`

Correct Answer is `=>` (C) `-ln (e^(-x) + 1) + C`

Q 2319156019

What is `int (d theta)/(sin^2 theta +2 cos^theta-1)` equal to?
NDA Paper 1 2008

(A)

`tan theta+c`

(B)

`cot theta+C`

(C)

`1/2 tan theta+C`

(D)

`1/2 cot theta+C`

Solution:

Let `I=int (d theta)/(sin^2theta+2 cos^2 theta-1)`

`=(d theta)/(1- cos^2 theta+2 cos^2 theta-1)=int (d theta)/(cos^2 theta)`

`=int sec^2 theta d theta=tan theta+C`

Correct Answer is `=>` (A) `tan theta+c`

Q 2379256116

What is `int sinx log(tanx) dx` equal to?
NDA Paper 1 2008

(A)

`cosx log tanx +log tan (x//2) + C`

(B)

`-cosx log tanx +log tan (x//2) + C`

(C)

`cosx log tanx +log cot (x//2) + C`

(D)

`-cosx log tanx +log cot (x//2) + C`

Solution:

`int sin x log (tan x) dx`

`-cos x log x(tan x)-int(-cos x) * 1/(tan x) * sec^2xdx`

`=-cos x log (tan x)+int 1/(sin x) dx`

`=-cos x log (tan x)+ int cosec x dx`

`cos x log x(tan x)+log (tan x/2)+C`

Correct Answer is `=>` (B) `-cosx log tanx +log tan (x//2) + C`

Q 2309256118

What is `int log(x + 1)dx` is equal to?
NDA Paper 1 2008

(A)

`x log (x+1)-x+C`

(B)

`(x+1)log(x+1)-x+C`

(C)

`1/(x+1)+C`

(D)

`(log(x+1))/(x+1)+C`

Solution:

Let `I=int log (x+1) dx`

Put `x + 1 => dx = dt`

`:. I = int 1*log t dt = t log t - int 1/t * t dt + C`

`= t log t - int 1 dt + C_1= t log t - t + C_1`

`=1 = (x + 1) log (x + 1)- x - 1 + C_1`

`= (x + 1) log (x + 1)- x + C`

where, `C =C_1 -1`

Correct Answer is `=>` (B) `(x+1)log(x+1)-x+C`

Q 2349356213

If `int(dx)/(f(x))= log {f(x)} ^2 + C`, then what is `f(x)` equal
to?
NDA Paper 1 2008

(A)

`2x+alpha`

(B)

`x+alpha`

(C)

`x/2+alpha`

(D)

`x^2+alpha`

Solution:

Let `f(x)=x/2+alpha`

`:. int (dx)/(x/2+alpha)=int(2 dx)/((x+2alpha))`

`=2*log (x + 2alpha)+ C_1`

`= log((x + 2alpha)^2 + C_1`

`=log{x/2+ alpha)^2 +log 2^ 2 + C_1`

`=log(x/2 + alpha)^2 + C`

where, `C =log 2 ^2 + C`
1

Correct Answer is `=>` (C) `x/2+alpha`

Q 2389356217

What is the value of `int(e^x(1+x))/(sin^2(xe^x)) dx` ?
NDA Paper 1 2007

(A)

`-e^x cot x + C`

(B)

`cos^2 (xe^x) + C`

(C)

`log sin(xe^x)+ C`

(D)

`-cot (xe^x)+ C`

Solution:

Let `I=int (e^x(1+x))/(sin^2(xe^x)) dx`

Put `xe^x=t` and `e^x(1+x)dx=dt`

`I=int(dl)/(sin^2t)-int cosec ^2 t dt`

`=- cot t + C =-cot (xe^x) + C`

Correct Answer is `=>` (D) `-cot (xe^x)+ C`