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Integral of cosxdx/3-sin^3x dx

The solution

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  1                      
  /                      
 |                       
 |  /cos(x)      3   \   
 |  |------ - sin (x)| dx
 |  \  3             /   
 |                       
/                        
0

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

Integral(cos(x)/3 - sin(x)^3, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \sin^{3}{\left(x \right)}\right)\, dx = - \int \sin^{3}{\left(x \right)}\, dx$
  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 $du$ :
      
      $\int \left(u^{2} - 1\right)\, du$
      
      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:
      
      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: $\frac{\cos^{3}{\left(x \right)}}{3}$
      
      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:
      
      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: $\frac{\cos^{3}{\left(x \right)}}{3}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      The result is: $\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$
  So, 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:
  
  $\int \frac{\cos{\left(x \right)}}{3}\, dx = \frac{\int \cos{\left(x \right)}\, dx}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $\frac{\sin{\left(x \right)}}{3}$
The result is: $\frac{\sin{\left(x \right)}}{3} - \frac{\cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

-2/3 - cos(1)^3/3 + sin(1)/3 + cos(1)

Numerical answer [src]

0.101549765720441

0.101549765720441

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

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How to use it?

Identical expressions

Similar expressions

Integral of cosxdx/3-sin^3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3