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sin(x)^ three + zero .5sin(2x)
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sin(x)^3-0.5sin(2x)
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Integral of sin(x)^3+0.5sin(2x) dx

The solution

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  p                        
  -                        
  2                        
  /                        
 |                         
 |  /   3      sin(2*x)\   
 |  |sin (x) + --------| dx
 |  \             2    /   
 |                         
/                          
0

$$\int\limits_{0}^{\frac{p}{2}} \left(\sin^{3}{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2}\right)\, dx$$

Integral(sin(x)^3 + sin(2*x)/2, (x, 0, p/2))

Detail solution

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

$\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

                          3/p\
                       cos |-|
11      /p\   cos(p)       \2/
-- - cos|-| - ------ + -------
12      \2/     4         3

$$\frac{\cos^{3}{\left(\frac{p}{2} \right)}}{3} - \cos{\left(\frac{p}{2} \right)} - \frac{\cos{\left(p \right)}}{4} + \frac{11}{12}$$

                          3/p\
                       cos |-|
11      /p\   cos(p)       \2/
-- - cos|-| - ------ + -------
12      \2/     4         3

$$\frac{\cos^{3}{\left(\frac{p}{2} \right)}}{3} - \cos{\left(\frac{p}{2} \right)} - \frac{\cos{\left(p \right)}}{4} + \frac{11}{12}$$

11/12 - cos(p/2) - cos(p)/4 + cos(p/2)^3/3

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

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Identical expressions

Similar expressions

Integral of sin(x)^3+0.5sin(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Method #1

Method #2