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Prove that int(0)^(pi//2) "cosec" (x-(pi...

Prove that `int_(0)^(pi//2) "cosec" (x-(pi)/(3))"cosec"(x-(pi)/(6)) dx = -2 log 3`

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int_(0)^(pi//2)log(cosx)dx=

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int_(0)^(pi//2)log(sinx)dx=

int_(0)^(pi) cos^(3) x dx =

int_(0)^(pi//2) (Sec x)/(Sec x + Cosec x) dx=

int_(0)^(2pi) x cos^(6) x dx=

Prove that int_(0)^(pi//2) tan^(-1)(sin x) .sin 2x = (pi)/(2)-1

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AAKASH SERIES-DEFINITE INTEGRALS-Exercise (Level-2)
  1. Prove that int(0)^(oo) (dx)/((x^(2)+a^(2))(x^(2)+b^(2)))=(pi)/(2ab(a+b...

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  2. Prove that int(a)^(oo) (dx)/(x^(4)sqrt(a^(2) + x^(2)))=(2-sqrt(2))/(3a...

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  3. Prove that int(0)^(pi//2) "cosec" (x-(pi)/(3))"cosec"(x-(pi)/(6)) dx =...

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  4. Show that int(0)^(oo) (dx)/([x+sqrt(x^(2)+1)]^(3)) = (3)/(8)

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  5. Show that int(0)^(pi)cos (2 cot^(-1) (sqrt((1-x)/(1+x))))dx = -(1)/(2)

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  6. Show that int(0)^(pi//2) (dx)/(3+4 six) = (1)/(sqrt(7)) log((4+sqrt(7)...

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  7. If int(log 2)^(x) (dy)/(sqrt(e^(y) -1)) = (pi)/(6), prove that x = log...

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  8. If int(log 2)^(x)(1)/(e^(x)-1)dx = log""(3)/(2), show that x = log 4

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  9. Show that int(0)^(1//2)(x sin^(-1)x)/(sqrt(1-x^(2)))dx = (1)/(2)-(sqrt...

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  10. Prove that int(0)^(pi//2) tan^(-1)(sin x) .sin 2x = (pi)/(2)-1

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  11. Show that (a) int(e)^(e^(2))(1)/(log x)dx = int(1)^(2)(e^(x))/(x)dx ...

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  12. Prove that int(0)^(1) log(sqrt(1-x)+sqrt(1+x))dx = (1)/(2) log 2 + (pi...

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  13. Show that int(0)^(pi//4) x cos x cos 3 x dx = (pi-3)/(16)

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  14. Prove that int(0)^(oo) e^(-2x)(sin 2x + cos 2 x) dx = (1)/(2)

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  15. Show that int(0)^(pi//2) (cos x)/((1+sin x)(2+sin x)) dx = log((4)/(3...

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  16. Show that int(1)^(7thsqrt(2))(1)/(x(2x^(7) + 1))dx=(1)/(7) log((6)/(5)...

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  17. Show that int(0)^(pi//2)(sin x cos x)/(cos^(2) x + 3 cos x + 2)dx = lo...

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  18. Show that int(0)^(1//3)(dx)/((1+x^(2))sqrt(1-x^(2))) = (pi)/(4sqrt(2))

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  19. Show that int(0)^(1)(dx)/((1+x^(2))sqrt(2+x^(2))) = (pi)/(6)

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  20. Show that int(0)^(1//2) (dx)/((1-2x^(2))sqrt(1-x^(2))) = (1)/(2) log(2...

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