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Identical expressions
(80cosx)*(sin^ three (x))
(80 co sinus of e of x) multiply by ( sinus of cubed (x))
(80 co sinus of e of x) multiply by ( sinus of to the power of three (x))
(80cosx)*(sin3(x))
80cosx*sin3x
(80cosx)*(sin³(x))
(80cosx)*(sin to the power of 3(x))
(80cosx)(sin^3(x))
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80cosxsin3x
80cosxsin^3x
(80cosx)*(sin^3(x))dx

Integral of (80cosx)*(sin^3(x)) dx

The solution

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 pi                     
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 4                      
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 |               3      
 |  80*cos(x)*sin (x) dx
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/                       
0

$$\int\limits_{0}^{\frac{\pi}{4}} \sin^{3}{\left(x \right)} 80 \cos{\left(x \right)}\, dx$$

Integral((80*cos(x))*sin(x)^3, (x, 0, pi/4))

Detail solution

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

Then let $du = \cos{\left(x \right)} dx$ and substitute $80 du$ :

$\int 80 u^{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u^{3}\, du = 80 \int u^{3}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{3}\, du = \frac{u^{4}}{4}$
  So, the result is: $20 u^{4}$
Now substitute $u$ back in:

$20 \sin^{4}{\left(x \right)}$
Add the constant of integration:

$20 \sin^{4}{\left(x \right)}+ \mathrm{constant}$

The answer is:

$20 \sin^{4}{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |              3                   4   
 | 80*cos(x)*sin (x) dx = C + 20*sin (x)
 |                                      
/

$$\int \sin^{3}{\left(x \right)} 80 \cos{\left(x \right)}\, dx = C + 20 \sin^{4}{\left(x \right)}$$

Simplify

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Plot the graph f(x)

The answer [src]

$$5$$

Numerical answer [src]

5.0

5.0

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

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Identical expressions

Integral of (80cosx)*(sin^3(x)) dx

Limits of integration:

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Piecewise:

The solution