Home
Class 12
MATHS
Evaluate: int0^pi(xsinx)/(1+cos^2x)\ dx...

Evaluate: `int_0^pi(xsinx)/(1+cos^2x)\ dx`

Text Solution

Verified by Experts

The correct Answer is:
`pi^(2)//4`
Let `I=int_(0)^(pi)(xsinxdx)/(1+cos^(2)x)`…………..1
or `I=int_(0)^(pi)((pi-x)sinx dx)/(1+cos^(2)x)`…………2
Adding 1 and 2 we get
`2I=piint_(0)^(pi)(sinxdx)/(1+cos^(2)x)`
or `I=-(pi)/2 int_(1)^(-1)(dt)/(1+t^(2))=-(pi)/2[tan^(-1)t]_(1)^(-1)`
[Putting `cosx=t, -sinx dx=dt`]
`=-1/2[tan^(-1)(-1)-tan^(-1)]=pi^(2)//4`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise SCQ_TYPE|113 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise MCQ_TYPE|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise SOLVED EXAMPLE_TYPE|20 Videos

  • CURVE TRACING

    CENGAGE ENGLISH|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE ENGLISH|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Evaluate: int_0^(pi/2)(x+sinx)/(1+cosx) dx

Evaluate: int_0^(pi//2)(sin x)/(1+cos^2x)dx

Evaluate: int_(0)^(pi) (xsinx)/(1+cos^(2)x)dx

int_(0)^(pi/2) (sinx)/(1+cos^(2)x)dx

Evaluate int _(0)^(pi)(x)/(1+ cos^(2)x)dx .

Evaluate int_(0)^(pi)(dx)/(1+3cos^(2)x)dx

Evaluate: int_-pi^pi ((2x)(1+sinx)/(1+cos^2x))dx

Evaluate: int_-pi^pi ((2x)(1+sinx)/(1+cos^2x))dx

Evaluate int_0^(2pi) sin^2x cos^4x dx

Evaluate: int_0^(pi//2)(xsinx cos\ x)/(sin^4x+cos^4x)dx