int (x^(2)(1-"ln"x))/(("ln"^(4)x-x^(4))d...

`int (x^(2)(1-"ln"x))/(("ln"^(4)x-x^(4))dx` is equal to

`(1)/(2) " In" ((x)/("In "x))-(1)/(4) "In"("In"^(2)x-x^(2))+C`

`(1)/(4)"In"(("In"x-x)/("In "x+x))-(1)/(2)tan^(-1)(("In "x)/(x))+C`

`(1)/(4) "In"(("In "x+x)/("In "x-x))+(1)/(2)tan^(-1)(("In "x)/(x))+C`

`(1)/(4) "In"(("In "x-x)/("In "x+x))+(1)/(2)tan^(-1)(("In "x)/(x))+C`

Text Solution

Verified by Experts

The correct Answer is:

Let `I=int(x^(2)(1-"In "x))/(("In "x)^(4)-x^(4))dx` Then,
`I=int(x^(2)(1-"In "x))/((("In "x)^(2)+x^(2))("In "x+x)("In "x-x))dx`
`rArrI=int((1-"In "x)/(x^(2)))/({(("In "x)/(x))^(2)+1}(("In "x)/(x)+1)(("In "x)/(x)-1))dx`
`rArrI=int(1)/((t^(2)+1)(t+1)(t-1))dt` , where `t=("In "x)/(x)`
`rArrI=int(1)/((t^(2)+1)(t^(2)-1))dt`
`rArrI=(1)/(2)int((1)/(t^(2)-1)-(1)/(t^(2)+1))dt=(1)/(4)"In"|(t-1)/(t+1)|-(1)/(2)tan^(-1)t+C`
`rArrI=(1)/(4)"In"|(("In"x)/(x)-1)/(("In "x)/(x)+1)|-(1)/(2)tan^(-1)(("In "x)/(x))+C`
`rArrI=(1)/(4)"In"|("In "x-x)/("In "x+x)|-(1)/(2)tan^(-1)(("In "x)/(x))+C`

Topper's Solved these Questions

INDEFINITE INTEGRALS
OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|3 Videos
INDEFINITE INTEGRALS
OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|61 Videos
INDEFINITE INTEGRALS
OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|30 Videos
INCREASING AND DECREASING FUNCTIONS
OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|18 Videos
INVERSE TRIGONOMETRIC FUNCTIONS
OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|30 Videos

Similar Questions

Explore conceptually related problems

int_(2-ln3)^(3+ln3)(ln(4+x))/(ln(4+x)+ln(9-x))dx is equal to :

int_(0)^(1//2)(1)/(1-x^(2))ln.(1-x)/(1+x)dx is equal to

The integral int_(2)^(4)(logx^(2))/(logx^(2)+log(36-12x+x^(2))) dx is equal to

int (f'(x))/( f(x) log(f(x)))dx is equal to

int(log(x+1)-logx)/(x(x+1))dx is equal to :

The value of the definite integral int e^(-x^4) (2+ln(x+sqrt(x^2+1))+5x^3-8x^4)dx is equal to

int_(0)^(infty)f(x+(1)/(x)) (ln x )/(x)dx is equal to:

int x ^(x)((ln x )^(2) +lnx+1/x) dx is equal to:

int ln(1+x)^(1+x)dx

Column I, a) int(e^(2x)-1)/(e^(2x)+1)dx is equal to b) int1/((e^x+e^(-x))^2)dx is equal to c) int(e^(-x))/(1+e^x)dx is equal to d) int1/(sqrt(1-e^(2x)))dx is equal to COLUMN II p) x-log[1+sqrt(1-e^(2x)]+c q) log(e^x+1)-x-e^(-x)+c r) log(e^(2x)+1)-x+c s) -1/(2(e^(2x)+1))+c

OBJECTIVE RD SHARMA ENGLISH-INDEFINITE INTEGRALS-Solved Example

"The integral " int(1+x-(1)/(x))e^(x+(1)/(x))dx " is equal to "
Text Solution
|
inte^(x^(4))(x+x^(3)+2x^(5))e^(x^(2))dx is equal to
Text Solution
|
int(sin(101x).sin^(99)x)dx equals
Text Solution
|
Suppose int(1-7cos^2x)/(sin^7xcos^2x)dx=(g(x))/(sin^7x)+c where C is a...
Text Solution
|
int(x^(2)+x)(x^(-8)+2x^(-9))^(1//10)dx is equal to
Text Solution
|
If int 2/(2-x)^2 ((2-x)/(2+x))^(1//3)\ dx = lambda ((2+x)/(2-x))^mu + ...
Text Solution
|
Let f(x) be a quadratic function such that f(0)=1 and int(f(x))/(x^2(x...
Text Solution
|
int (x dx)/(sqrt(1+x^(2)+sqrt((1+x^(2))^(3))) is equal to
Text Solution
|
int(e^(x)(x-1)(x-lnx))/(x^(2))dx is equal to
Text Solution
|
intx^(27)(1+x+x^(2))^(6)(6x^(2)+5x+4)dx=
Text Solution
|
Let f (x) be a quadratic function such that f (0) =1 and f(-1)=4, if...
Text Solution
|
Evaluate int(e^(tan^(-1)x))/((1+x^(2)))[(sec^(-1)sqrt(1+x^(2)))^(2)+"c...
Text Solution
|
int (x^2(xsec^2x+tanx))/(xtanx+1)^2
Text Solution
|
int (mx^(m+2n-1)-nx^(n-1))/(x^(2m+2n)+2x^(m+n)+1)dx is equal to
Text Solution
|
Evaluate int(dx)/(tanx+cotx+secx+cosecx).
Text Solution
|
int (x^(2)(1-"ln"x))/(("ln"^(4)x-x^(4))dx is equal to
Text Solution
|
The integral int (2x^(12)+5x^(9))/((x^(5)+x^(3)+1)^(3))dx is equal to
Text Solution
|
If int(dx)/(cos^(3)xsqrt(sin2x))=a(tan^(2)x+b)sqrt(tanx)+C, then
Text Solution
|
"If"int(dx)/(x^(3)(1+x^(6))^(2/3))=xf(x)(1+x^(6))^(1/3)+C where, C is ...
Text Solution
|
int(dx)/((1+sqrt(x))sqrt((x-x^(2)))) is equal to
Text Solution
|