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Evaluate: int0^(pi//2)1/((a^2cos^2x+b^2s...

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

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Verified by Experts

We have given,
`int_0^(pi//2)1/((a^2cos^2x+b^2sin^2x)^2)dx`
`I=int_0^(pi//2)1/((a^2cos^2x+b^2sin^2x)^2)dx`
`I=int_0^(pi//2)(sec^4x)/((a^2+b^2tan^2x)^2)dx`
`I=int_0^(pi//2)((1+tan^2x)sec^2x)/((a^2+b^2tan^2x)^2)dx`
`I=int_0^(oo)(1+t^2x)/((a^2+b^2t^2)^2)dt`
On solving we get,
`I=(1/(b^2))int_0^(oo)(1/(a^2+b^2t^2))dt+(1/(b^2))int_0^(oo)((b^2-a^2)/(a^2+b^2t^2))dt`
...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. If In=int0^(pi//4)tan^("n")x dx , prove that In+I(n-2)=1/(n-1)dot

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  2. If In=int0^(pi//4)tan^n x dx , show that 1/(I2+I4),1/(I3+I5),1/(I4+...

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  3. Evaluate: int0^(pi//2)1/((a^2cos^2x+b^2sin^2x)^2)dx

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  4. Evaluate: int0^1x(tan^(-1)x)^2dx

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  5. Evaluate: int0^a1/((x^2+a^2))dx (ii) int0^oo(x^2)/((a^2+x^2)^(5//2))dx

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  6. Evaluate: int0^1xsqrt((1-x^2)/(1+x^2))dx

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  7. Evaluate the following integrals: int0^a(sin)^(-1)sqrt(x/(a+x))dx

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

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  9. Evaluate the following integrals: int0^(pi//4)(sin^2xcos^2x)/((sin...

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  10. Evaluate the following integrals: int0^(pi//2)(tanx)/(1+m^2tan^2x)d...

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  11. Evaluate: int0^1|5x-3|dx

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  12. Evaluate: (i) \int(-1)^1f(x)dx ,\where\, f(x)={1-2x ,xlt=0 ;1+2x ,xgt=...

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  13. int (1/e)^e |logx|dx =

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  14. If a >0, find int0^(3a)|x^2-a^2|dxdot

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  15. If [dot] denotes the greatest integer function, then find the valu...

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  16. Evaluate: int0^3[x]dx

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  17. int0^1d/(dx){sin^(-1)((2x)/(1+x^2))}dx is equal to 0 (b) pi (c) ...

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  18. Evaluate the following integrals: int(-a)^a(xe^(x^2))/(1+x^2)dx

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  19. Evaluate the following integrals: int0^(pi//4)|cos2x|dx

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  20. Evaluate the following integrals: int-5^0f(x)dx ,w h e r ef(x)=|x|+|x+...

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