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Prove that int(0)^(1) ("arc cosx")^(n...

Prove that
`int_(0)^(1) ("arc cosx")^(n) dx = n ((pi)/(2))^(n-1) - n (n - 1) int_(0)^(1) ("arc cos x")^(n-1) dx (n gt 1)`

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Verified by Experts

The correct Answer is:
`u = ("arc cos x) ^(n-1)` the second time.
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IA MARON-THE DEFINITE INTEGRAL -6 . 6 (Integration by Parts. Reduction Formulas)
  1. Compute the integral int (1)^(0) I n ^(3) x dx

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  2. Compute the integral int(0)^(pi^(2)/4) sin sqrt(x) dx

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  3. Compute the integral I = int(0)^(1) ("arc sin x")/(sqrt(1 - x^(2)))dx

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  4. int(0)^(pi//2) x^(2) sin x " " dx=

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  5. Compute the integral I(n) = int(0)^(a) (a^(2) - x^(2))^(n) dx , where...

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  6. Using the result of the preceding problem obtain the following formula...

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  7. Compute the integral H(m) = int(0)^(pi//2) sin^(m) x dx = int(0)^(p...

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  8. Compute the integral I = int(0)^(x) x sin^(m) x dx (m is natur...

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  9. Compute the integral I (n) = int(0)^(1) x^(m) (I n x)^(n) dx , m gt 0,...

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  10. Compute the integral I(m,n) = int(0)^(1) x^(m) (! - x)^(n) dx , where...

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  11. Compute the integrals : int(0)^(1) " arc tan " sqrt(x) dx

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  12. Compute the integrals : int (x - 1)e^(-x) dx

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  13. Compute the integrals : int(pi//4)^(pi//3) (x dx)/( sin^(2) x)

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  14. Compute the integrals : int(0)^(1) x "arc tan x dx

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  15. Compute the integrals : int(0)^(1) x I n (1 + x^(2)) dx

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  16. int(0)^(pi//4) log (1+tan x) dx =?

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  17. Compute the integrals : int (0) ^(pi//2) " sin In 2 x arc tan " ...

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  18. Compute the integrals : int(1) ^(15) "arc tan " sqrt(sqrt(x) - 1)...

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  19. Prove that int(0)^(1) ("arc cosx")^(n) dx = n ((pi)/(2))^(n-1) - n ...

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  20. Prove that if f'' is continuous on [a,b] then the following formula...

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