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Prove that int(0)^(1) log(sqrt(1-x)+sqrt...

Prove that `int_(0)^(1) log(sqrt(1-x)+sqrt(1+x))dx = (1)/(2) log 2 + (pi)/(4) - (1)/(2)`

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AAKASH SERIES-DEFINITE INTEGRALS-Exercise (Level-2)
  1. Prove that int(0)^(pi//2) tan^(-1)(sin x) .sin 2x = (pi)/(2)-1

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

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

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

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

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

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

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

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

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

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

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  12. int(0)^(log5)(e^(x)sqrt(e^(x)-1))/(e^(x)+3)dx=

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  13. Show that int(0)^(pi//2) (cos^(2) theta)/(cos^(2) theta + 4 sin^(2)...

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  14. Show that int(0)^(pi//4) sqrt(tan theta) d theta = (1)/(sqrt(2)) log(s...

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  15. Show that int(0)^(pi//4)(dx)/(cos^(4)x - cos^(2)x sin^(2) x + sin^(4)x...

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  16. Show that int(0)^(pi//2)(sinxcos x)/(1+sin^(4)x)dx = (pi)/(8)

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  17. Prove that int(0)^(pi//2)(cos^(2) x sin x)/(sqrt(1+cos^(2)x))dx = (sqr...

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  18. Show that int(0)^(oo) (dx)/((x+sqrt(1+x^(2)))^(n))=(n)/(n^(2)-1), (n i...

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  19. int(0)^(pi//4)[sqrt(tan x) + sqrt(cot x)] = dx

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  20. Show that int(0)^(pi//2)(sqrt(tan x)+sqrt(cot x))dx = pisqrt(2)

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