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Transform the definite integral int(0)^(...

Transform the definite integral `int_(0)^(2 pi) f(x) cos x dx` by the substitution sin x = t

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The correct Answer is:
`int_(0)^(1) f` (arc sin t) `dt + int _(1)^(-1) (pi - "arc sin" t) dt + int_(-1)^(0) f (2 pi + "arc sin " t) dt`
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IA MARON-THE DEFINITE INTEGRAL -6 . 4 (Changing the Variable in a Definite Integral)
  1. Prove that the integral int(0)^(pi) (sin 2 k x)/(sin x) dx equal...

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  2. Compute the integral int(1//2)^(sqrt(3//2)) (dx)/(x sqrt(1 - x^(2)))

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  3. Prove that the function L(x) defined on the interval (0,infty) by the ...

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  4. Transform the integral int(0)^(3) (x-2)^(2) dx by the substitution (x...

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  5. Compute the integrals I = int(0)^(1)(dx)/(1 + sqrt(x))

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  6. Compute the integrals I = int(0)^(5) (dx)/( 2 x + sqrt(3 x + 1))

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  7. Compute the integrals I = int(pi//4)^(pi//3) (dx)/( 1 - sin x)

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  8. Compute the integrals I = int(0)^(1) sqrt(2 x - x^(2)) dx

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  9. Compute the integrals I = int(0)^(pi//4) (sin x + cos x)/( 3 + si...

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  10. Compute the integrals I = int(0)^(a) x^(2) sqrt((a - x)/( a + x))...

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  11. Compute the integrals I = int(0)^(2a) sqrt(2 a x - x^(2)) dx

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  12. Compute the integrals I = int(-1)^(1) (dx)/(1 + x^(2))

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  13. Applying a suitable change of the variable, find the following defini...

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  14. Consider the integral int(-2)^(2) (dx)/(4 + x^(2)) . It is easy to co...

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  15. Consider the integral I = int(0)^(2pi) (dx)/( 5 - 2 cos x) . Making th...

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  16. Make sure that a formal change of the variable t = x^((2)/(5)) leads ...

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  17. Is is possible to make the substitution x = sec t in the integral I = ...

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  18. Given the integral int(0)^(1) sqrt(1 - x^(2)) dx. Made the substitut...

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  19. Prove the equality int(-a)^(a) f(x) dx = int(0)^(a) [f(x) + f(-x)]...

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  20. Transform the definite integral int(0)^(2 pi) f(x) cos x dx by the su...

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