Step by step solution to "Evaluate the following integrals. Include absolute values. ∫01yln4(y2+1)y2+1dy"

geduiwelh

Answered question

2021-10-13

Evaluate the following integrals. Include absolute values.

\int_{0}^{1} \frac{y \ln^{4} (y^{2} + 1)}{y^{2} + 1} d y

Answer & Explanation

Faiza Fuller

Skilled2021-10-14Added 108 answers

Step 1
To Determine: Evaluate the following integrals. Include absolute values.
Given: we have an integral $\int_{0}^{1} \frac{y \ln^{4} (y^{2} + 1)}{y^{2} + 1} d y$
let us consider $u = y^{2} \Rightarrow d u = 2 y d y \Rightarrow \frac{d u}{2} = y d y$
y=0 then u = 0
y=1 then u =1
then the integral becomes
$\int_{0}^{1} \frac{\ln^{4} (u + 1)}{2 (u + 1)} d u = \frac{1}{2} \int_{0}^{1} \frac{\ln^{4} (u + 1)}{(u + 1)} d u$
now again let $v = u + 1 \Rightarrow d v = d u$
the limits becomes
u=0 then v = 1
u=1 then v = 2
so the integral becomes
$\frac{1}{2} \int_{1}^{2} \frac{\ln^{4} v}{v} d v$
let $w = \ln v \Rightarrow d w = \frac{1}{v} d v$
limits becomes
v=1 then w = 0
v=2 then $w = \ln 2$
so the integral becomes
Step 2
$\frac{1}{2} \int_{0}^{\ln 2} w^{4} d w$
on integrating we have
$\frac{1}{2} \int_{0}^{\ln 2} w^{4} d w = \frac{1}{2} {[\frac{w^{5}}{5}]}_{0}^{\ln 2}$
$= \frac{\ln^{5} 2}{10}$
Answer:the integral $\int_{0}^{1} \frac{y \ln^{4} (y^{2} + 1)}{(y^{2} + 1)} d y = \frac{\ln^{5} 2}{10}$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Integral Calculus

Didn't find what you were looking for?