Home
Class 12
MATHS
Evaluate: int0^(sqrt(3))1/(1+x^2)sin^(-1...

Evaluate: `int_0^(sqrt(3))1/(1+x^2)sin^(-1)((2x)/(1+x^2))dx`

Text Solution

Verified by Experts

For`0ltxlt1, "sin"^(-1)(2x)/(1+x^(2))=2tan^(-1)x`
For `xgt1, "sin"^(-1)(2x)/(1+x^(2))="sin"^(-1)(2/x)/(1+1/(x^(2)))=2tan^(-1)(1//x)=2cot^(-1)x`
Hence the given integral is
`int_(0)^(sqrt(3))("sin"^(-1)((2x)/(1+x^(2))))/((1+x^(2))) dx=int_(0)^(1)2/(1+x^(2))(tan^(-1)x)dx+int_(1)^(sqrt(3))(2cot^(-1)x)/(1+x^(2))dx`
`=[(tan^(-1)x)^(2)]_(0)^(1)-[(cot^(-)x)^(2)]_(1)^(sqrt(3))=(pi^(2))/16-((pi^(2))/36-(pi^(2))/16)`
`=(pi^(2))/8-(pi^(2))/36=(7pi^(2))/72`
Promotional Banner

Similar Questions

Explore conceptually related problems

Evaluate: int_0^1 1/(sqrt(1-x^2))sin^(-1)(2xsqrt(1-x^2))dxdot

Evaluate: int_0^1 2xsqrt(1+x^2) dx

Evaluate: int_0^(1/(sqrt(2)))(sin^(-1)x)/((1-x^2)sqrt(1-x^2))dx

Evaluate: int_0^1 xsqrt(1+x^2) dx

Evaluate: int_(0)^(1) dx/sqrt(4-x^2)

int(2sin^(-1)x)/(sqrt(1-x^(2)))dx

Evaluate: int_(0)^(1/sqrt2)(sin^-1x)/((1-x^(2))^(3/2) dx.

Evaluate: int_(sqrt(2))^(sqrt(2)+1)((x^2-1))/((x^2+1)^2)dx

Evaluate int_(0)^(1)(x^2/(1+x^2))dx

Evaluate: int1/(x^2sqrt(1+x^2))dx