Juan Hewlett

2022-01-03

Evaluate the indefinite integral.
$\int {\left(4x+5\right)}^{9}dx$

temnimam2

Step 1
To Determine: evaluate the indefinite integral.
Given: we have $\int {\left(4x+5\right)}^{9}dx$
Explanation: we have $\int {\left(4x+5\right)}^{9}dx$
let us consider $u=4x+5⇒du=4dx$
then the integral becomes
$\int {\left(4x+5\right)}^{9}dx$
$=\frac{1}{4}\int {u}^{9}du$
$=\frac{1}{4}\frac{{u}^{10}}{10}$
Step 2
now putting the value of u then we have
$\int {\left(4x+5\right)}^{9}dx=\frac{1}{4}\frac{{\left(4x+5\right)}^{10}}{10}$
$=\frac{{\left(4x+5\right)}^{10}}{40}+c$

Given:
$\int {\left(4x+5\right)}^{9}dx$
$=\frac{1}{4}\int {u}^{9}du$
Now we calculate:
$\int {u}^{9}du$
$=\frac{{u}^{10}}{10}$
We substitute the already calculated integrals:
$\frac{1}{4}\int {u}^{9}du$
$=\frac{{u}^{10}}{40}$
$=\frac{{\left(4x+5\right)}^{10}}{40}$
$=\frac{{\left(4x+5\right)}^{10}}{40}+C$

karton

$\int \left(4x+5{\right)}^{9}dx\phantom{\rule{0ex}{0ex}}\text{Need to make a replacement}\phantom{\rule{0ex}{0ex}}\int \frac{{t}^{9}}{4}dt\phantom{\rule{0ex}{0ex}}\text{use properties}\phantom{\rule{0ex}{0ex}}\frac{1}{4}\ast \int {t}^{9}dt\phantom{\rule{0ex}{0ex}}\frac{1}{4}\ast \frac{{t}^{10}}{10}\phantom{\rule{0ex}{0ex}}\text{Substitute back}\phantom{\rule{0ex}{0ex}}\frac{1}{4}\ast \frac{\left(4x+5{\right)}^{10}}{10}\phantom{\rule{0ex}{0ex}}\frac{\left(4x+5{\right)}^{10}}{40}\phantom{\rule{0ex}{0ex}}\text{Solution:}\phantom{\rule{0ex}{0ex}}\frac{\left(4x+5{\right)}^{10}}{40}+C$