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Identical expressions
sin^10xcos^9x
sinus of to the power of 10x co sinus of e of to the power of 9x
sin10xcos9x
sin^10xcos⁹x
sin^10xcos^9xdx

Integral of sin^10xcos^9x dx

The solution

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 pi                    
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 2                     
  /                    
 |                     
 |     10       9      
 |  sin  (x)*cos (x) dx
 |                     
/                      
0

$$\int\limits_{0}^{\frac{\pi}{2}} \sin^{10}{\left(x \right)} \cos^{9}{\left(x \right)}\, dx$$

Integral(sin(x)^10*cos(x)^9, (x, 0, pi/2))

Detail solution

Rewrite the integrand:

$\sin^{10}{\left(x \right)} \cos^{9}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{4} \sin^{10}{\left(x \right)} \cos{\left(x \right)}$
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 $du$ :
  
  $\int \left(u^{18} - 4 u^{16} + 6 u^{14} - 4 u^{12} + u^{10}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 4 u^{16}\right)\, du = - 4 \int u^{16}\, du$
      
      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: $- \frac{4 u^{17}}{17}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 6 u^{14}\, du = 6 \int u^{14}\, du$
      
      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: $\frac{2 u^{15}}{5}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 4 u^{12}\right)\, du = - 4 \int u^{12}\, du$
      
      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: $- \frac{4 u^{13}}{13}$
    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: $\frac{u^{19}}{19} - \frac{4 u^{17}}{17} + \frac{2 u^{15}}{5} - \frac{4 u^{13}}{13} + \frac{u^{11}}{11}$
  Now substitute $u$ back in:
  
  $\frac{\sin^{19}{\left(x \right)}}{19} - \frac{4 \sin^{17}{\left(x \right)}}{17} + \frac{2 \sin^{15}{\left(x \right)}}{5} - \frac{4 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{4} \sin^{10}{\left(x \right)} \cos{\left(x \right)} = \sin^{18}{\left(x \right)} \cos{\left(x \right)} - 4 \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 6 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 4 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + \sin^{10}{\left(x \right)} \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{18}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $- \frac{4 \sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 6 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 6 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $\frac{2 \sin^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $- \frac{4 \sin^{13}{\left(x \right)}}{13}$
  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: $\frac{\sin^{19}{\left(x \right)}}{19} - \frac{4 \sin^{17}{\left(x \right)}}{17} + \frac{2 \sin^{15}{\left(x \right)}}{5} - \frac{4 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{4} \sin^{10}{\left(x \right)} \cos{\left(x \right)} = \sin^{18}{\left(x \right)} \cos{\left(x \right)} - 4 \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 6 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 4 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + \sin^{10}{\left(x \right)} \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{18}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $- \frac{4 \sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 6 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 6 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $\frac{2 \sin^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $- \frac{4 \sin^{13}{\left(x \right)}}{13}$
  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: $\frac{\sin^{19}{\left(x \right)}}{19} - \frac{4 \sin^{17}{\left(x \right)}}{17} + \frac{2 \sin^{15}{\left(x \right)}}{5} - \frac{4 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$
Now simplify:

$\frac{\left(12155 \sin^{8}{\left(x \right)} - 54340 \sin^{6}{\left(x \right)} + 92378 \sin^{4}{\left(x \right)} - 71060 \sin^{2}{\left(x \right)} + 20995\right) \sin^{11}{\left(x \right)}}{230945}$
Add the constant of integration:

$\frac{\left(12155 \sin^{8}{\left(x \right)} - 54340 \sin^{6}{\left(x \right)} + 92378 \sin^{4}{\left(x \right)} - 71060 \sin^{2}{\left(x \right)} + 20995\right) \sin^{11}{\left(x \right)}}{230945}+ \mathrm{constant}$

The answer is:

$\frac{\left(12155 \sin^{8}{\left(x \right)} - 54340 \sin^{6}{\left(x \right)} + 92378 \sin^{4}{\left(x \right)} - 71060 \sin^{2}{\left(x \right)} + 20995\right) \sin^{11}{\left(x \right)}}{230945}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                    
 |                                13           17         11         19           15   
 |    10       9             4*sin  (x)   4*sin  (x)   sin  (x)   sin  (x)   2*sin  (x)
 | sin  (x)*cos (x) dx = C - ---------- - ---------- + -------- + -------- + ----------
 |                               13           17          11         19          5     
/

$$\int \sin^{10}{\left(x \right)} \cos^{9}{\left(x \right)}\, dx = C + \frac{\sin^{19}{\left(x \right)}}{19} - \frac{4 \sin^{17}{\left(x \right)}}{17} + \frac{2 \sin^{15}{\left(x \right)}}{5} - \frac{4 \sin^{13}{\left(x \right)}}{13} + \frac{\sin^{11}{\left(x \right)}}{11}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

 128  
------
230945

$$\frac{128}{230945}$$

 128  
------
230945

$$\frac{128}{230945}$$

128/230945

Numerical answer [src]

0.000554244517092814

0.000554244517092814

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

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How to use it?

Identical expressions

Integral of sin^10xcos^9x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3