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

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

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Let I=`int_0^(pi//2)(cos^2x)/(cos^2x+4sin^2x)dx`
⇒I=`int_0^(pi//2)(cos^2x)/(cos^2x+4(1-cos^2x)dx`
​ ⇒`I=−1/3​int_0^(pi//2)(4−3cos^2x-4)/(4−3cos^2x)​dx`
⇒`I=−1/3​int_0^(pi//2)(4−3cos^2x)/(4−3cos^2x)​dx+1/3​int_0^(pi//2)(4)/(4−3cos^2x)​dx`
⇒`I=−1/3int ​_0^(π/2)​​1dx+1/3​int_0^(pi//2)(4sec^2x)/(4sec^2x−3)​dx`
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate: int0^(pi//2)(xsinxcosx)/(sin^4x+cos^4x)dx

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  19. Evaluate the following integral: int0^(pi//2)(dx)/(acosx+bsinx)x

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  20. Evaluate the following integral: int0^pi(sinx)/(sinx+cosx)dx

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