<s>[INST] Find the integral $\int \:x^{-\frac{1}{2}}\cdot \left(1+x^{\frac{1}{4}}\right)^{-10} dx$ Help me find the integral. I think we have to somehow replace apply.
$$\int \:x^{-\frac{1}{2}}\cdot \left(1+x^{\frac{1}{4}}\right)^{-10} dx =\int \frac{1}{\sqrt{x} (1+x^{\frac{1}{4}})^{10}} dx $$
 [/INST] You may just perform the change of variable $x=u^4$, to get
$$
\begin{align}
\int \:x^{-\frac{1}{2}}\cdot \left(1+x^{\frac{1}{4}}\right)^{-10} dx&=4\int \:u^{-2}\cdot \left(1+u\right)^{-10} u^3du\\\\
&=4\int \frac{1}{(1+u)^9} du-4\int  \frac{1}{(1+u)^{10}}du\\\\
&=-\frac{1}{2 (1+u)^8}+\frac{4}{9 (1+u)^9}+C\\\\
&=-\frac{1}{2 \left(1+x^{1/4}\right)^9}+\frac{4}{9 \left(1+x^{1/4}\right)^9}+C.
\end{align}
$$
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