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Identical expressions
cos^7xsin^10x
co sinus of e of to the power of 7x sinus of to the power of 10x
cos7xsin10x
cos⁷xsin^10x
cos^7xsin^10xdx

Integral of cos^7xsin^10x dx

The solution

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  1                    
  /                    
 |                     
 |     7       10      
 |  cos (x)*sin  (x) dx
 |                     
/                      
0

$$\int\limits_{0}^{1} \sin^{10}{\left(x \right)} \cos^{7}{\left(x \right)}\, dx$$

Integral(cos(x)^7*sin(x)^10, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{10}{\left(x \right)} \cos^{7}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{3} \sin^{10}{\left(x \right)} \cos{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \sin{\left(x \right)}$ .
  
  Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
  
  $\int \left(- u^{16} + 3 u^{14} - 3 u^{12} + u^{10}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{16}\right)\, du = - \int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      So, the result is: $- \frac{u^{17}}{17}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 u^{14}\, du = 3 \int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      So, the result is: $\frac{u^{15}}{5}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 3 u^{12}\right)\, du = - 3 \int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      So, the result is: $- \frac{3 u^{13}}{13}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
    The result is: $- \frac{u^{17}}{17} + \frac{u^{15}}{5} - \frac{3 u^{13}}{13} + \frac{u^{11}}{11}$
  Now substitute $u$ back in:
  
  $- \frac{\sin^{17}{\left(x \right)}}{17} + \frac{\sin^{15}{\left(x \right)}}{5} - \frac{3 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{3} \sin^{10}{\left(x \right)} \cos{\left(x \right)} = - \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + \sin^{10}{\left(x \right)} \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{17}{\left(x \right)}}{17}$
    So, the result is: $- \frac{\sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 3 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{15}{\left(x \right)}}{15}$
    So, the result is: $\frac{\sin^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 3 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{13}{\left(x \right)}}{13}$
    So, the result is: $- \frac{3 \sin^{13}{\left(x \right)}}{13}$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{10}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{11}{\left(x \right)}}{11}$
  The result is: $- \frac{\sin^{17}{\left(x \right)}}{17} + \frac{\sin^{15}{\left(x \right)}}{5} - \frac{3 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{3} \sin^{10}{\left(x \right)} \cos{\left(x \right)} = - \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + \sin^{10}{\left(x \right)} \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{17}{\left(x \right)}}{17}$
    So, the result is: $- \frac{\sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 3 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{15}{\left(x \right)}}{15}$
    So, the result is: $\frac{\sin^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 3 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{13}{\left(x \right)}}{13}$
    So, the result is: $- \frac{3 \sin^{13}{\left(x \right)}}{13}$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{10}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{11}{\left(x \right)}}{11}$
  The result is: $- \frac{\sin^{17}{\left(x \right)}}{17} + \frac{\sin^{15}{\left(x \right)}}{5} - \frac{3 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$
Now simplify:

$\frac{\left(- 715 \sin^{6}{\left(x \right)} + 2431 \sin^{4}{\left(x \right)} - 2805 \sin^{2}{\left(x \right)} + 1105\right) \sin^{11}{\left(x \right)}}{12155}$
Add the constant of integration:

$\frac{\left(- 715 \sin^{6}{\left(x \right)} + 2431 \sin^{4}{\left(x \right)} - 2805 \sin^{2}{\left(x \right)} + 1105\right) \sin^{11}{\left(x \right)}}{12155}+ \mathrm{constant}$

The answer is:

$\frac{\left(- 715 \sin^{6}{\left(x \right)} + 2431 \sin^{4}{\left(x \right)} - 2805 \sin^{2}{\left(x \right)} + 1105\right) \sin^{11}{\left(x \right)}}{12155}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                     
 |                                13         17         15         11   
 |    7       10             3*sin  (x)   sin  (x)   sin  (x)   sin  (x)
 | cos (x)*sin  (x) dx = C - ---------- - -------- + -------- + --------
 |                               13          17         5          11   
/

$$\int \sin^{10}{\left(x \right)} \cos^{7}{\left(x \right)}\, dx = C - \frac{\sin^{17}{\left(x \right)}}{17} + \frac{\sin^{15}{\left(x \right)}}{5} - \frac{3 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       13         17         15         11   
  3*sin  (1)   sin  (1)   sin  (1)   sin  (1)
- ---------- - -------- + -------- + --------
      13          17         5          11

$$- \frac{3 \sin^{13}{\left(1 \right)}}{13} - \frac{\sin^{17}{\left(1 \right)}}{17} + \frac{\sin^{11}{\left(1 \right)}}{11} + \frac{\sin^{15}{\left(1 \right)}}{5}$$

       13         17         15         11   
  3*sin  (1)   sin  (1)   sin  (1)   sin  (1)
- ---------- - -------- + -------- + --------
      13          17         5          11

$$- \frac{3 \sin^{13}{\left(1 \right)}}{13} - \frac{\sin^{17}{\left(1 \right)}}{17} + \frac{\sin^{11}{\left(1 \right)}}{11} + \frac{\sin^{15}{\left(1 \right)}}{5}$$

-3*sin(1)^13/13 - sin(1)^17/17 + sin(1)^15/5 + sin(1)^11/11

Numerical answer [src]

0.00103319601406097

0.00103319601406097

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

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How to use it?

Identical expressions

Integral of cos^7xsin^10x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3