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Solving integrals/ 2sin^3x*cosx

Integral of 2sin^3x*cosx dx

The solution

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  1                    
  /                    
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 |       3             
 |  2*sin (x)*cos(x) dx
 |                     
/                      
0

$$\int\limits_{0}^{1} 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$$

Integral(2*sin(x)^3*cos(x), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{3}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{4}{\left(x \right)}}{4}$
  Method #2
  1. Rewrite the integrand:
    
    $\sin^{3}{\left(x \right)} \cos{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$
  2. Let $u = - \cos^{2}{\left(x \right)}$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \left(\frac{u}{2} + \frac{1}{2}\right)\, du$
    1. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u}{2}\, du = \frac{\int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, du = \frac{u}{2}$
      
      The result is: $\frac{u^{2}}{4} + \frac{u}{2}$
    Now substitute $u$ back in:
    
    $\frac{\cos^{4}{\left(x \right)}}{4} - \frac{\cos^{2}{\left(x \right)}}{2}$
So, the result is: $\frac{\sin^{4}{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{\sin^{4}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sin^{4}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                              4   
 |      3                    sin (x)
 | 2*sin (x)*cos(x) dx = C + -------
 |                              2   
/

$$\int 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{4}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   4   
sin (1)
-------
   2

$$\frac{\sin^{4}{\left(1 \right)}}{2}$$

   4   
sin (1)
-------
   2

$$\frac{\sin^{4}{\left(1 \right)}}{2}$$

Упростить

Numerical answer [src]

0.25068398283281

0.25068398283281

The graph

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

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Identical expressions

Integral of 2sin^3x*cosx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2