Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of √(x^4-2x^3+x^2)
Integral of cos^6x
Integral of (3x+4)²
Integral of sin^2tdt
Identical expressions
sin^ four (x)cos(x)
sinus of to the power of 4(x) co sinus of e of (x)
sinus of to the power of four (x) co sinus of e of (x)
sin4(x)cos(x)
sin4xcosx
sin⁴(x)cos(x)
sin^4xcosx
sin^4(x)cos(x)dx
Similar expressions
sin^4x*cos(x)
sin^4(x)cosx

Integral of sin^4(x)cos(x) dx

The solution

You have entered [src]

 pi                  
 --                  
 3                   
  /                  
 |                   
 |     4             
 |  sin (x)*cos(x) dx
 |                   
/                    
pi                   
--                   
6

$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$$

Integral(sin(x)^4*cos(x), (x, pi/6, pi/3))

Detail solution

Let $u = \sin{\left(x \right)}$ .

Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :

$\int u^{4}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int u^{4}\, du = \frac{u^{5}}{5}$
Now substitute $u$ back in:

$\frac{\sin^{5}{\left(x \right)}}{5}$
Add the constant of integration:

$\frac{\sin^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\sin^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                            5   
 |    4                    sin (x)
 | sin (x)*cos(x) dx = C + -------
 |                            5   
/

$$\int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{5}{\left(x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

            ___
   1    9*\/ 3 
- --- + -------
  160     160

$$- \frac{1}{160} + \frac{9 \sqrt{3}}{160}$$

            ___
   1    9*\/ 3 
- --- + -------
  160     160

$$- \frac{1}{160} + \frac{9 \sqrt{3}}{160}$$

-1/160 + 9*sqrt(3)/160

Numerical answer [src]

0.0911778579257494

0.0911778579257494

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin^4(x)cos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution