Integrate sin(4x)^4cos(4x)

\int\sin\left(4x\right)^4\cos\left(4x\right)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

$\frac{1}{20}\sin\left(4x\right)^{5}+C_0$

Step by step solution

Problem

$\int\sin\left(4x\right)^4\cos\left(4x\right)dx$
1

Solve the integral $\int\frac{\sin\left(4x\right)^{\left(u-1\right)}\sin\left(4x\right)^4}{-4}du$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=\sin\left(4x\right) \\ du=4\cos\left(4x\right)dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{4\cos\left(4x\right)}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{u^4}{4}du$
4

Taking the constant out of the integral

$\frac{1}{4}\int u^4du$
5

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

$\frac{1}{4}\cdot\frac{u^{5}}{5}$
6

Substitute $u$ back for it's value, $\sin\left(4x\right)$

$\frac{1}{4}\cdot\frac{\sin\left(4x\right)^{5}}{5}$
7

Simplify the fraction

$\frac{1}{20}\sin\left(4x\right)^{5}$
8

Add the constant of integration

$\frac{1}{20}\sin\left(4x\right)^{5}+C_0$

Answer

$\frac{1}{20}\sin\left(4x\right)^{5}+C_0$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.34 seconds

Views:

154