Question

Integrate the following with respect to x.

`("e"^(3logx))/(x^4 + 1)`

Answer 1

`("e"^(3logx))/(x^4 + 1) = ("e"^(log x^3))/(x^4 + 1)`

`x^3/(x^4 + 1)`

Let x⁴ + 1 = f(x)

Then 4x³ = f'(x)

So `int ("e"^(3logx))/(x^4 + 1)  "d"x = int x^3/(x^4 + 1)  "d"x`

= `1/4 int (4x^3)/(x^4 + 1)  "d"x`

= `1/4 int ("f'"(x))/"f'(x)  "d"x`

= `1/4 log |"f"(x)| + "c"`

= `1/4 log|x^4 + 1| + "c"`