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Integral of d{x}:
Integral of (1+x)/x
Integral of 3^x*e^x
Integral of 1/√(1-x²)
Integral of e^(x+1)
Derivative of:
cos^7(x)
Identical expressions
cos^ seven (x)
co sinus of e of to the power of 7(x)
co sinus of e of to the power of seven (x)
cos7(x)
cos7x
cos⁷(x)
cos^7x
cos^7(x)dx
Similar expressions
cos^7x*sinx*
cos^7x
cos^7x*sinx

Solving integrals/ cos^7(x)

Integral of cos^7(x) dx

The solution

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  1           
  /           
 |            
 |     7      
 |  cos (x) dx
 |            
/             
0

$$\int\limits_{0}^{1} \cos^{7}{\left(x \right)}\, dx$$

Integral(cos(x)^7, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\cos^{7}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{3} \cos{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{3} \cos{\left(x \right)} = - \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{\sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 3 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    So, the result is: $\frac{3 \sin^{5}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 3 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
    So, the result is: $- \sin^{3}{\left(x \right)}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{3 \sin^{5}{\left(x \right)}}{5} - \sin^{3}{\left(x \right)} + \sin{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{3} \cos{\left(x \right)} = - \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{\sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 3 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    So, the result is: $\frac{3 \sin^{5}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 3 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
    So, the result is: $- \sin^{3}{\left(x \right)}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{3 \sin^{5}{\left(x \right)}}{5} - \sin^{3}{\left(x \right)} + \sin{\left(x \right)}$
Now simplify:

$\left(- \frac{\sin^{6}{\left(x \right)}}{7} + \frac{3 \sin^{4}{\left(x \right)}}{5} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$
Add the constant of integration:

$\left(- \frac{\sin^{6}{\left(x \right)}}{7} + \frac{3 \sin^{4}{\left(x \right)}}{5} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\left(- \frac{\sin^{6}{\left(x \right)}}{7} + \frac{3 \sin^{4}{\left(x \right)}}{5} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                               7           5            
 |    7                3      sin (x)   3*sin (x)         
 | cos (x) dx = C - sin (x) - ------- + --------- + sin(x)
 |                               7          5             
/

$$-{{\sin ^7x}\over{7}}+{{3\,\sin ^5x}\over{5}}-\sin ^3x+\sin x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

               7           5            
     3      sin (1)   3*sin (1)         
- sin (1) - ------- + --------- + sin(1)
               7          5

$$-{{5\,\sin ^71-21\,\sin ^51+35\,\sin ^31-35\,\sin 1}\over{35}}$$

               7           5            
     3      sin (1)   3*sin (1)         
- sin (1) - ------- + --------- + sin(1)
               7          5

$$- \sin^{3}{\left(1 \right)} - \frac{\sin^{7}{\left(1 \right)}}{7} + \frac{3 \sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)}$$

Simplify

Numerical answer [src]

0.456104465133679

0.456104465133679

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos^7(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2