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sinx+sin^2x+sin^3x
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Similar expressions
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Solving integrals/ sinx+sin^2x+sin^3x

Integral of sinx+sin^2x+sin^3x dx

The solution

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  1                                
  /                                
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 |  /            2         3   \   
 |  \sin(x) + sin (x) + sin (x)/ dx
 |                                 
/                                  
0

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

Integral(sin(x) + sin(x)^2 + sin(x)^3, (x, 0, 1))

Detail solution

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

$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} - \frac{7 \cos{\left(x \right)}}{4} + \frac{\cos{\left(3 x \right)}}{12}$
Add the constant of integration:

$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} - \frac{7 \cos{\left(x \right)}}{4} + \frac{\cos{\left(3 x \right)}}{12}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} - \frac{7 \cos{\left(x \right)}}{4} + \frac{\cos{\left(3 x \right)}}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                   3                   
13              cos (1)   cos(1)*sin(1)
-- - 2*cos(1) + ------- - -------------
6                  3            2

$$- 2 \cos{\left(1 \right)} - \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{3}{\left(1 \right)}}{3} + \frac{13}{6}$$

                   3                   
13              cos (1)   cos(1)*sin(1)
-- - 2*cos(1) + ------- - -------------
6                  3            2

$$- 2 \cos{\left(1 \right)} - \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{3}{\left(1 \right)}}{3} + \frac{13}{6}$$

13/6 - 2*cos(1) + cos(1)^3/3 - cos(1)*sin(1)/2

Numerical answer [src]

0.911313899974298

0.911313899974298

The graph

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

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Identical expressions

Similar expressions

Integral of sinx+sin^2x+sin^3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3