dedica66em

2021-12-13

Determine the following integrals.
$\frac{y}{{\left({y}^{2}+1\right)}^{\frac{5}{4}}}dy$

Juan Spiller

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

servidopolisxv

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

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