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Compute the integrals : (a) int(0)^(0...

Compute the integrals :
(a) ` int_(0)^(0) x^(2) sqrt(a^(2) - x^(2)) dx`
(b) ` int_(1)^(sqrt(x)) (dx)/(sqrt((1 + x^(2))^(3)))`

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Verified by Experts

The correct Answer is:
(a) `(pi a^(4))/(16) ` (substitution x = a sin t)
(b) `(sqrt(3)-sqrt(2))/(2)` (substitution x = tan t)
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IA MARON-THE DEFINITE INTEGRAL -6 . 4 (Changing the Variable in a Definite Integral)
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  2. The value of the integral overset(4)underset(2)int (sqrt(x^(2)-4))/(x^...

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  3. Compute the integrals : (a) int(0)^(0) x^(2) sqrt(a^(2) - x^(2)) dx...

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  6. Compute the integrals (a) int(1)^(sqrt(3)) (sqrt(1 + x^(2)))/(x^(2)...

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  7. The value of the integralunderset(0)overset(pi)int (x sin x)/(1+cos^(2...

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

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

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

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

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

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

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

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