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int(x^3dx)/(sqrt(1+x^2))= a(1+x^(2))^(3/...

`int(x^3dx)/(sqrt(1+x^2))= a(1+x^(2))^(3//2) + bsqrt(1+x^(2)) + C`, then

A

`a = 1/3, b = 1`

B

`a = (-1)/(3), b = 1`

C

`a = (-1)/(3), b = - 1`

D

`a = 1/3, b = -1`

Text Solution

Verified by Experts

The correct Answer is:
A
Let ` I = int(x^(3))/(sqrt(1+x^(2))) dx = a(1+x^(2))^(3//2) + bsqrt(1+x^(2)) + C`
`:' I = int(x^(3))/(sqrt(1+x^(2))) dx = int (x^(2).x)/(sqrt(1+x^(2))) dx`
Put ` 1+x^(2) = t^(2)`
`rArr 2x dx = 2tdt`
`:. I = int (t(t^(2) - 1))/(t) dt = (t^(3))/(3) - t + C`
`= 1/3 (1+x^(2))^(3//2) - sqrt(1+x^(2))+C`
`:. a = 1/3` and `b = - 1`
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