Question

How do you find the antiderivative of `int (6x)/(x^2-7)^(1/9)dx` from `[3,4]`?

Answer 1

The integral is approximately equal to `17.546`.

Explanation:

The first thing to do with definite integrals is to make sure that they're in fact definite and not improper.

The integral `(6x)/(x^2 -7)^(1/9)` is continuous on `[3,4]`. We are dealing with a definite integral.

I think we should consider a u-substitution to integrate. Let `u = x^2 - 7`. Then `du = 2xdx` and `dx = (du)/(2x)`. Furthermore, the new bounds of integration become `2` to `9` because we will now be working in `u`.

`=>int_2^9 (6x)/u^(1/9) * (du)/(2x)`

`=>int_2^9 3/u^(1/9)`

`=>3int_2^9 u^(-1/9)`

`=>3[9/8u^(8/9)]_2^9`

`=>3[9/8(9)^(8/9) - 9/8(2)^(8/9)]`

`~~17.546`

Hopefully this helps!