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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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Evaluate: int_0^(pi//2)logtanx\ dx

int_(0)^(pi)log(1+cosx)dx=-pi(log2)

Prove that int_0^(pi//2) sin 2x log tanx dx=0

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A DAS GUPTA-Properties and Application of definite Integrals-EXERCISE
  1. Prove that int(0)^(pi//2)(2logsinx-logsin2x)dx=(pi)/(2)(log2).

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  2. int(0)^(pi//2)(sqrt(tanx)+sqrt(cotx))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. int(0)^(pi//2)log(tanx+cotx)dx=pi(log2)

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  5. int(0)^(oo)(x logx)/((1+x^(2))^(2)) dx=

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  6. int(pi//6)^(pi//3)(1)/((1+sqrt(tanx)))dx=(pi)/(12)

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  7. Q. int0^pi(e^(cos^2x)( cos^3(2n+1) x dx, n in I

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  8. int(0)^(2pi) (sin 2 theta)/(a-b cos 2 theta )d theta =

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  9. Evaluate int0^1 log(1+x)/(1+x^2)dx

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  10. Prove that: y=int(1/8)^(sin^2x)sin^(-1)sqrt(t)dt+int(1/8)^(cos^2x)cos^...

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  11. Evaluate the following: int0^(pi/2) |cosx-sinx|dx

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  12. Evaluate the following:int-2^2(|x|+|x-1|)dx

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  13. Prove that inta^b{[x]+[-x]}dx=a-b,where [x]is the greatest integer lex...

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  14. Statement-I int0^9[sqrtx]dx=13, Statement-II int0^(n^2) [sqrt x]dx=(n(...

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  15. Show that: int0^x[x]dx=[x]([x]-1)/2+[x](x-[x]), where [x] denotes the ...

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  16. The value of the integral overset(100pi)underset(0)int sqrt(1-cos2x)" ...

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  17. Evaluate the following : int(0)^(pi)(dx)/(1+sinx)

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  18. Evaluate the following:int0^(10pi)|cosx|dx

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  19. Evaluate int(0)^(npi+t)(|cosx|+|sinx|)dx, where n epsilonN and t epsil...

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  20. Show that: inta^bf(x)dx=int(a+c)^(b+c)f(x-c)dx and hence show that int...

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