Question

Evaluate the indefinite integral. (use c for the constant of integration.) int (x**(7))/(1 + x**(16)) dx

Answer 1

$\displaystyle\int\frac{x^7}{1+x^{16}}\,\mathrm dx$

Notice that $x^{16}=(x^8)^2$, so let's make the substitution $y=x^8$, so that $\mathrm dy=8x^7\,\mathrm dx$. Then the above can be written as

$\displaystyle\frac18\int\frac{\mathrm dy}{1+y^2}=\frac18\tan^{-1}y+C=\frac18\tan^{-1}x^8+C$