Question

How do you integrate for partial fractions: ∫arctg²x/x²+1 dx ∫e^√x/√x dx ? thanks! good nights

Answer 1

Partial fractions aren't even required for these. For the first problem, let `u = arctanx`. Then `du = 1/(1 + x^2)dx` and `dx = du(1 + x^2)`.

`I = int u^2/(x^2 + 1) * du(x^2 + 1)`

`I = int u^2du`

`I = 1/3u^3 + C`

`I = 1/3arctan^3x + C`

Now for the second, we let `u = sqrt(x)`. This means that `du = 1/(2sqrt(x))dx` and `dx = 2sqrt(x)du`.

`I = int e^u/sqrt(x) * 2sqrt(x)du`

`I = 2int e^u du`

`I = 2e^u + C`

`I = 2e^sqrt(x) + C`

Hopefully this helps!