burkinaval1b

2022-01-03

Use the Change of Variables Formula to evaluate the definite integral.
${\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx$

Foreckije

Step 1
Using Change of variables we have to evaluate the definite integral.
${\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx$
Step 2
${\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx$
Let us substitute ${x}^{2}+9$ by t
So, when x=0, t=9
and when x=4, t=25
Differentiating ${x}^{2}+9=t⇒2xdx=dt⇒xdx=\frac{dt}{2}$
So we can write the integral ${\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx$ as $\frac{1}{2}{\int }_{9}^{25}\sqrt{t}dt$
Now $\frac{1}{2}{\int }_{9}^{25}\sqrt{t}dt=\frac{{t}^{\frac{3}{2}}}{3}{\mid }_{9}^{25}=\frac{125}{3}-\frac{27}{3}=\frac{98}{3}$
So, ${\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx=\frac{98}{3}$

Corgnatiui

${\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx$
Let us put the expression 2 * x under the differential sign, i.e.:
$2xdx=d\left({x}^{2}\right),t={x}^{2}$
Then the original integral can be written as follows:
$\int \frac{\sqrt{t+9}}{2}dt$
We make a change of variables:
t=x+9
Then:
$\int \frac{\sqrt{x+9}}{2}dx=\int \frac{\sqrt{x+9}}{2}\cdot d\left(x+9\right)=\int \frac{\sqrt{t}}{2}dt=\frac{{t}^{\frac{3}{2}}}{3}$
Back to x:
$\frac{{\left(x+9\right)}^{\frac{3}{2}}}{3}$
$\frac{{\left(x+9\right)}^{\frac{3}{2}}}{3}+C$
To write the final answer, it remains to substitute ${x}^{2}$ instead of t.
$\frac{{\left({x}^{2}+9\right)}^{\frac{3}{2}}}{3}+C$
Lets

karton

$\phantom{\rule{0ex}{0ex}}\text{Step 1}\phantom{\rule{0ex}{0ex}}\text{Given}\phantom{\rule{0ex}{0ex}}{\int }_{0}^{4}x\sqrt{{x}^{2}+9}dx\phantom{\rule{0ex}{0ex}}\int x\sqrt{{x}^{2}+9}dx\phantom{\rule{0ex}{0ex}}\int \frac{1}{2}×\sqrt{t}dt\phantom{\rule{0ex}{0ex}}\frac{1}{2}×\int \sqrt{t}dt\phantom{\rule{0ex}{0ex}}\text{Transform the expression}\phantom{\rule{0ex}{0ex}}\frac{1}{2}×\int {t}^{\frac{1}{2}}dt\phantom{\rule{0ex}{0ex}}\text{Step 2}\phantom{\rule{0ex}{0ex}}\text{Evaluate the integral}\phantom{\rule{0ex}{0ex}}\frac{1}{2}×\frac{2t\sqrt{t}}{3}\phantom{\rule{0ex}{0ex}}\frac{1}{2}×\frac{2\left({x}^{2}+9\right)\sqrt{{x}^{2}+9}}{3}\phantom{\rule{0ex}{0ex}}\frac{\left({x}^{2}+9\right)\sqrt{{x}^{2}+9}}{3}\phantom{\rule{0ex}{0ex}}\frac{\left({x}^{2}+9\right)\sqrt{{x}^{2}+9}}{3}{|}_{0}^{4}\phantom{\rule{0ex}{0ex}}\text{Calculate the expression}\phantom{\rule{0ex}{0ex}}\frac{\left({4}^{2}+9\right)\sqrt{{4}^{2}+9}}{3}-\frac{\left({0}^{2}+9\right)\sqrt{{0}^{2}+9}}{3}\phantom{\rule{0ex}{0ex}}\text{Answer:}\phantom{\rule{0ex}{0ex}}\frac{98}{3}$