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Prove that: int0^(pi//2)logsinx\ dx=\ in...

Prove that: `int_0^(pi//2)logsinx\ dx=\ int_0^(pi//2)logcosx\ dx=-pi/2log2`

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`I=int_0^(pi//2)logsinx\ dx ---(1)`
`I=int_0^(pi//2)logsin(pi/2 -x) dx`
`=int_0^(pi//2)log cosx dx ---(2)`
Adding 1 and 2,
`2I=int_0^(pi//2)(logsinx+log cosx) dx`
`I=1/2 int_0^(pi//2)logsinx(sin2x/2)dx`
`2x=z=>2dx=dz`
`x=pi/2` and `z=pi` and `x=0, z=0` ...
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Prove that int_0^(pi//2) sin 2x log tanx dx=0

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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate: int(-1)^(3//2)|xsinpix|dx

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  2. Evaluate: int0^(2pi)cos^5x\ dx

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  3. Prove that: int0^(pi//2)logsinx\ dx=\ int0^(pi//2)logcosx\ dx=-pi/2log...

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  4. Evaluate: int0^pilog(1+cosx)dx

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  5. Evaluate the following integral: int0^(pi//2)1/(1+tanx)

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  6. Evaluate the following integral: int0^(pi//2)(sqrt(cotx))/(sqrt(cotx\ ...

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  7. Evaluate the following integral: int0^a1/(x+sqrt(a^2-x^2))dx

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  8. Evaluate the following integral: int0^(pi//2)1/(1+cotx)dx

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  9. Evaluate the following integral: int0^(pi//2)(sin^(3//2)\ )/(sin^(3//2...

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  10. Evaluate the following integral: int0^(pi//2)1/(1+tanx)dx

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

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

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  13. Evaluate the following integral:int0^pi(xtanx)/(secx\ cos e c\ x)dx

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  14. Evaluate the following integral:int0^pixsin^3x\ dx

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  15. Evaluate the following integral:int0^pi(xsinx)/(1+sinx)dx

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  16. Evaluate the following integral:int0^(pi//2)(tan^7x)/(tan^7x+cot^7x)dx

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  17. Evaluate the following integral:int0^(pi//2)x\ s in xcos^2x\ dx

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  18. Evaluate the following integral:int(-pi//2)^(pi//2)sin^3x\ dx

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  19. Evaluate the following integral:int(-1)^1log((2-x)/(2+x))dx\

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  20. Evaluate the following integral:int0^pilog(1-cos x)dx

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