Question

`\int\limits {(2x^7)/(x^4+4)} \, dx`

Answer 1

`\displaystyle \int (2x^7)/(x^4 + 4) \, dx`

Let
`u=x^4 + 4` and
`du=4x^3\,dx`. Then

`\displaystyle \int (2x^7)/(x^4 + 4) \, dx = \frac14 \int \frac{2(u-4)}u \, du \\\\ ~~~~~~~~ = \frac12 \int \left(1 - \frac4u\right) \, dx \\\\ ~~~~~~~~ = \frac12 \left(u - 4\ln|u|\right) + C \\\\ ~~~~~~~~ = \frac12 \left(x^4 +4 - 4\ln\left|x^4+4\right|\right) + C \\\\ ~~~~~~~~ = \boxed{\frac{x^4}2 - \ln\left(x^4+4\right)^2 + C}`