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Prove that: int0^(2pi)(xsin^(2n)x)/(...

Prove that: `int_0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx`= `pi^2`

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To prove that \[ I = \int_0^{2\pi} \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx = \pi^2, \] we will follow a systematic approach. ...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. For x >0,l e tf(x)=int1^x((log)e t)/(1+t)dtdot Find the function...

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  2. Evaluate the following integrals: (1-35) int0^(pi//2)(sin^n x)/(sin...

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  3. Prove that: int0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx= pi^2

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  4. Prove that: int0^(pi//2)log|tanx+cotx|dx=pi(log)e2

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  5. Evaluate: int(-2)^2|xcospix|dx

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  6. Evaluate: int(pi//4)^(pi//4)log(sinx+cosx)dx

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  7. Evaluate: int0^pixlogsinxdx

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  8. If int0^pi 1/(a+bcosx)dx=pi/sqrt(a^2-b^2), then int0^pi 1/(a+bcosx)^2d...

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  9. int0^1sqrt((1-x)/(1+x))dx= (pi)/2 (b) (pi)/2-1 (c) (pi)/2+1 (d) ...

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  10. int0^(pi//2)(cosx)/((2+sinx)(1+sinx))dx equals (a)log(2/3) (b) l...

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  11. int0^pi(xtanx)/(secx+cosx)dxi s (pi^2)/4 (b) (pi^2)/2 (c) (3pi^2)/2...

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  12. int0^2 1/(1+tanx)dx is equal to (pi^)/4 (b) (pi^)/3 (c) (pi^)/2 ...

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  13. int(-pi//2)^(pi//2)"sin"|x|dx is equal to 1 (b) 2 (c) -1 (d) ...

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  14. The value of the integral int0^oox/((1+x)(1+x^2))dx is (pi)/2 (b) ...

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  15. int0^3(3x+1)/(x^2+9)dx = (pi)/(12)+log(2sqrt(2)) (b) (pi)/2+log(2sqr...

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  16. I10=int0^(pi/2)x^(10)sinx dx then I10+90I8 is (A) 10(pi/2)^6 (B) 10(p...

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  17. Evaluate:int0^(pi//2)(cosx)/(1+cosx+sinx)dx

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  18. Evaluate: int0^(pi//2)1/(3+2cosx)dx

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  19. Evaluate: int(pi//4)^(pi//2)cos2xlogsinxdx

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  20. integrate int0^(2pi) e^x . sin (pi/4 + x/2) dx

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