Determine the following integrals. y(y2+1)54dy

Step 1
Our Aim is to evaluate the integral given below:−
${\displaystyle \int \frac{y}{{(y^2+1)}^{\frac{5}{4}}}dy}$...(i)
Step 2
Consider the integral given by equation-(i), we have:−
${\displaystyle \int \frac{y}{{(y^2+1)}^{\frac{5}{4}}}dy-\left(i\right)}$
For the integrand ${\displaystyle \frac{y}{{(y^2+1)}^{\frac{5}{4}}}}$, we will substitute ${\displaystyle t=y^2+1}$ and dt=2ydy
$=\frac{1}{2}\int \frac{1}{t^{\frac{5}{4}}}dt$
Since, the integral of $\frac{1}{t^{\frac{5}{4}}}is\frac{-4}{\sqrt[4]{t}}$
$=\frac{-2}{\sqrt[4]{t}}+constant$
Answer $=\frac{-2}{\sqrt[4]{y^2+1}}+constant$

Given:
${\displaystyle \int \frac{y}{{(y^2+1)}^{\frac{5}{4}}}dy}$
$\int \frac{1}{2t^{\frac{5}{4}}}dt$
Use properties of integrals
$\frac{1}{2}\ast \int \frac{1}{t^{\frac{5}{4}}}dt$
$\frac{1}{2}\ast (-\frac{4}{\sqrt[4]{t}})$
Substitute back
$\frac{1}{2}\ast (-\frac{4}{\sqrt[4]{y^2+1}})$
$-\frac{2}{\sqrt[4]{y^2+1}}$
Result:
$-\frac{2}{\sqrt[4]{y^2+1}}+C$