Integral of sin^3xdx dx

The solution

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

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

Detail solution

Rewrite the integrand:

$\sin^{3}{\left(x \right)} 1 = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$
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:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    1. 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:
    
    $\int \left(- \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      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:
    
    $\int \left(- \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      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: $\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$
Now simplify:

$\frac{\left(\cos{\left(2 x \right)} - 5\right) \cos{\left(x \right)}}{6}$
Add the constant of integration:

$\frac{\left(\cos{\left(2 x \right)} - 5\right) \cos{\left(x \right)}}{6}+ \mathrm{constant}$

The answer is:

$\frac{\left(\cos{\left(2 x \right)} - 5\right) \cos{\left(x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                3   
 |    3                        cos (x)
 | sin (x)*1 dx = C - cos(x) + -------
 |                                3   
/

$$\int \sin^{3}{\left(x \right)} 1\, dx = \frac{\cos^{3}{\left(x \right)}}{3} + C - \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                3   
2            cos (1)
- - cos(1) + -------
3               3

$${{\cos ^31-3\,\cos 1}\over{3}}+{{2}\over{3}}$$

                3   
2            cos (1)
- - cos(1) + -------
3               3

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

Simplify

Numerical answer [src]

0.178940562548858

0.178940562548858

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sin^3xdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3