## Problem Statement

Find the indefinite integral:

$$
\int \frac{\sqrt{1+\sqrt{x}}}{x \sqrt[4]{x^{3}}} d x
$$
## Solution

$$
\int \frac{\sqrt{1+\sqrt{x}}}{x \sqrt[4]{x^{3}}} d x=\int x^{-\frac{7}{4}}\left(1+x^{\frac{1}{2}}\right)^{\frac{1}{2}} d x=
$$

Under the integral is a binomial differential $x^{m}\left(a+b x^{n}\right)^{p}$, from which

$$
m=-\frac{7}{4} ; n=\frac{1}{2} ; p=\frac{1}{2}
$$

Since $\frac{m+1}{n}+p=\frac{-\frac{7}{4}+1}{\frac{1}{2}}+\frac{1}{2}=-\frac{3}{2}+\frac{1}{2}=-1$ is an integer, we use the substitution:

$$
a x^{-n}+b=z^{s} \text {, where } s \text { is the denominator of the fraction } p \text {. }
$$

That is, in our case, the substitution has the form:

$$
\begin{aligned}
& x^{-\frac{1}{2}}+1=z^{2} \\
& x=\left(z^{2}-1\right)^{-2} \\
& d x=-2\left(z^{2}-1\right)^{-3} \cdot 2 z d z=-4 z\left(z^{2}-1\right)^{-3} d z
\end{aligned}
$$

We get:

$$
\begin{aligned}
& =\int\left(\left(z^{2}-1\right)^{-2}\right)^{-\frac{7}{4}}\left(1+\left(\left(z^{2}-1\right)^{-2}\right)^{\frac{1}{2}}\right)^{\frac{1}{2}}(-4 z)\left(z^{2}-1\right)^{-3} d z= \\
& =\int\left(z^{2}-1\right)^{\frac{7}{2}}\left(1+\left(z^{2}-1\right)^{-1}\right)^{\frac{1}{2}}(-4 z)\left(z^{2}-1\right)^{-3} d z= \\
& =\int\left(z^{2}-1\right)^{\frac{1}{2}}\left(1+\frac{1}{z^{2}-1}\right)^{\frac{1}{2}}(-4 z) d z=\int\left(z^{2}-1\right)^{\frac{1}{2}}\left(\frac{z^{2}}{z^{2}-1}\right)^{\frac{1}{2}}(-4 z) d z= \\
& =\int\left(z^{2}-1\right)^{\frac{1}{2}} \cdot \frac{z}{\left(z^{2}-1\right)^{\frac{1}{2}}}(-4 z) d z=-4 \int z^{2} d z=-\frac{4 z^{3}}{3}+C=
\end{aligned}
$$

$$
=-\frac{4\left(x^{-\frac{1}{2}}+1\right)^{\frac{3}{2}}}{3}+C=-\frac{4 \sqrt{\left(\frac{1}{\sqrt{x}}+1\right)^{3}}}{3}+C=-\frac{4 \sqrt{(1+\sqrt{x})^{3}}}{3 \sqrt[4]{x}}+C
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+13-1$ » Categories: Kuznetsov's Problem Book Integrals Problem $13 \mid$ Integrals

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## Problem Kuznetsov Integrals 13-2

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