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Similar expressions
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Solving integrals/ (sin(2*x))*(cos(x)^(2))

Integral of (sin(2x))(cos(x)^(2)) dx

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2                2                2                                                        
1   cos (1)*cos(2)   sin (1)*sin(2)   cos (1)*sin(2)   cos(1)*cos(2)*sin(1)   cos(1)*sin(1)*sin(2)
- - -------------- - -------------- + -------------- - -------------------- - --------------------
2         2                4                4                   2                      4

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

       2                2                2                                                        
1   cos (1)*cos(2)   sin (1)*sin(2)   cos (1)*sin(2)   cos(1)*cos(2)*sin(1)   cos(1)*sin(1)*sin(2)
- - -------------- - -------------- + -------------- - -------------------- - --------------------
2         2                4                4                   2                      4

1/2 - cos(1)^2*cos(2)/2 - sin(1)^2*sin(2)/4 + cos(1)^2*sin(2)/4 - cos(1)*cos(2)*sin(1)/2 - cos(1)*sin(1)*sin(2)/4

Numerical answer [src]

0.457389435440761

0.457389435440761

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (sin(2x))(cos(x)^(2)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (sin(2*x))*(cos(x)^(2)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of (sin(2x))(cos(x)^(2)) dx