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Evaluate: int0^(pi//2)(xsinxcosx)/(sin^4...

Evaluate: `int_0^(pi//2)(xsinxcosx)/(sin^4x+cos^4x)dx`

A

`(pi^(2))/2`

B

`(pi^(2))/4`

C

`(pi^(2))/8`

D

`(pi^(2))/16`

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AI Generated Solution

To evaluate the integral \[ I = \int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx, \] we can use the property of definite integrals that states: ...
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STATEMENT-1 : int_(0)^((pi)/(2))(xsinxcosx)/(sin^(4)x+cos^(4)x)dx=(pi^(2))/(32) and STATEMENT-2 : int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx

RD SHARMA ENGLISH-DEFINITE INTEGRALS-All Questions
  1. Evaluate: int0^(pi//2)(sin x)/(1+cos^2x)dx

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  2. Evaluate: int0^(pi//2)1/(4sin^2x+5cos^2x)dx

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  3. Evaluate: int0^(pi//2)(xsinxcosx)/(sin^4x+cos^4x)dx

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  4. Evaluate: int0^(pi//4)(sin\ 2x)/(cos^4x+sin^4x)dx

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  5. Evaluate: int0^(pi//2)(cos^2x)/(cos^2x+4sin^2x)dx

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  6. Evaluate the following integral: int2^4x/(x^2+1)dx

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  7. Evaluate the following integral: int1^2 1/(x(1+logx)^2)dx

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  8. Evaluate the following integral: int1^2(3x)/(9x^2-1)dx

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  9. Evaluate the following integral: int0^a x/(sqrt(a^2+x^2))dx

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  10. Evaluate the following integral: int0^1x e^x^2dx

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  11. Evaluate the following integral: int0^1(2x)/(1+x^4)dx

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  12. Evaluate the following integral: int1^2 1/(x(1+logx)^2)dx

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  13. Evaluate the following integral: int0^(pi//2)1/(5cos x+3sinx)dx

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  14. Evaluate the following integral: int0^1(e^x)/(1+e^(2x))dx

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  15. Evaluate the following integral: int1^3(cos(logx))/x dx

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  16. Evaluate the following integral: int0^asqrt(a^2-x^2)dx

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  17. Evaluate the following integral: int0^(pi//2)sqrt(sinvarphi)cos^5varph...

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  18. Evaluate the following integral: int0^(pi//2)(sintheta)/(sqrt(1+costhe...

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  19. Evaluate the following integral: int0^1(sqrt(tan^(-1)x))/(1+x^2)dx

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  20. Evaluate the following integral: int0^1tan^(-1)\ ((2x)/(1-x^2))dx

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