Integral of x(x+2)^0.5

\int x\sqrt{x+2}dx

Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

e
π
ln
log
lim
d/dx
d/dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Answer

$-\frac{4}{3}\sqrt{\left(2+x\right)^{3}}+\frac{2}{5}\sqrt{\left(2+x\right)^{5}}+C_0$

Step by step solution

Problem

$\int x\sqrt{x+2}dx$
1

Solve the integral $\int x\sqrt{2+x}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=2+x \\ du=dx\end{matrix}$
2

Rewriting $x$ in terms of $u$

$x=u-2$
3

Substituting $u$, $dx$ and $x$ in the integral

$\int\sqrt{u}\left(u-2\right)du$
4

Multiplying polynomials $\sqrt{u}$ and $u+-2$

$\int\left(\sqrt{u^{3}}-2\sqrt{u}\right)du$
5

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int-2\sqrt{u}du+\int\sqrt{u^{3}}du$
6

Taking the constant out of the integral

$\int\sqrt{u^{3}}du-2\int\sqrt{u}du$
7

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\int\sqrt{u^{3}}du-2\cdot \frac{2}{3}\sqrt{u^{3}}$
8

Substitute $u$ back for it's value, $2+x$

$\int\sqrt{u^{3}}du-\frac{4}{3}\sqrt{\left(2+x\right)^{3}}$
9

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{2}{5}\sqrt{u^{5}}-\frac{4}{3}\sqrt{\left(2+x\right)^{3}}$
10

Substitute $u$ back for it's value, $2+x$

$\frac{2}{5}\sqrt{\left(2+x\right)^{5}}-\frac{4}{3}\sqrt{\left(2+x\right)^{3}}$
11

Add the constant of integration

$-\frac{4}{3}\sqrt{\left(2+x\right)^{3}}+\frac{2}{5}\sqrt{\left(2+x\right)^{5}}+C_0$

Answer

$-\frac{4}{3}\sqrt{\left(2+x\right)^{3}}+\frac{2}{5}\sqrt{\left(2+x\right)^{5}}+C_0$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.61 seconds

Views:

77