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Integral of d{x}:
Integral of sin^7(x)*cos^2(x)
Integral of x^3-3x^2
Integral of 10x^4+30x^2–24x^5
Integral of ((pi-x)/2)*cos(nx)
Identical expressions
sin^ seven (x)*cos^ two (x)
sinus of to the power of 7(x) multiply by co sinus of e of squared (x)
sinus of to the power of seven (x) multiply by co sinus of e of to the power of two (x)
sin7(x)*cos2(x)
sin7x*cos2x
sin⁷(x)*cos²(x)
sin to the power of 7(x)*cos to the power of 2(x)
sin^7(x)cos^2(x)
sin7(x)cos2(x)
sin7xcos2x
sin^7xcos^2x
sin^7(x)*cos^2(x)dx

Solving integrals/ sin^7(x)*cos^2(x)

Integral of sin^7(x)*cos^2(x) dx

The solution

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    32                     
     /                     
    |                      
    |       7       2      
    |    sin (x)*cos (x) dx
    |                      
   /                       
     ___                   
12*\/ 3

$$\int\limits_{12 \sqrt{3}}^{32} \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

Integral(sin(x)^7*cos(x)^2, (x, 12*sqrt(3), 32))

Detail solution

Rewrite the integrand:

$\sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
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 $du$ :
  
  $\int \left(u^{8} - 3 u^{6} + 3 u^{4} - u^{2}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 3 u^{6}\right)\, du = - 3 \int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $- \frac{3 u^{7}}{7}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 u^{4}\, du = 3 \int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      So, the result is: $\frac{3 u^{5}}{5}$
    1. 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: $- \frac{u^{3}}{3}$
    The result is: $\frac{u^{9}}{9} - \frac{3 u^{7}}{7} + \frac{3 u^{5}}{5} - \frac{u^{3}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{8}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(x \right)} \cos^{8}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{8}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{8}\right)\, du = - \int u^{8}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
      
      So, the result is: $- \frac{u^{9}}{9}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{9}{\left(x \right)}}{9}$
    So, the result is: $\frac{\cos^{9}{\left(x \right)}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{6}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{6}\right)\, du = - \int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $- \frac{u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{3 \cos^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{4}\right)\, du = - \int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      So, the result is: $- \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{5}{\left(x \right)}}{5}$
    So, the result is: $\frac{3 \cos^{5}{\left(x \right)}}{5}$
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int u^{2}\, du$
    1. 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:
    
    $- \frac{\cos^{3}{\left(x \right)}}{3}$
  The result is: $\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{8}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(x \right)} \cos^{8}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{8}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{8}\right)\, du = - \int u^{8}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
      
      So, the result is: $- \frac{u^{9}}{9}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{9}{\left(x \right)}}{9}$
    So, the result is: $\frac{\cos^{9}{\left(x \right)}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{6}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{6}\right)\, du = - \int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $- \frac{u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{3 \cos^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{4}\right)\, du = - \int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      So, the result is: $- \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{5}{\left(x \right)}}{5}$
    So, the result is: $\frac{3 \cos^{5}{\left(x \right)}}{5}$
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int u^{2}\, du$
    1. 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:
    
    $- \frac{\cos^{3}{\left(x \right)}}{3}$
  The result is: $\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$
Now simplify:

$\frac{\left(35 \cos^{6}{\left(x \right)} - 135 \cos^{4}{\left(x \right)} + 189 \cos^{2}{\left(x \right)} - 105\right) \cos^{3}{\left(x \right)}}{315}$
Add the constant of integration:

$\frac{\left(35 \cos^{6}{\left(x \right)} - 135 \cos^{4}{\left(x \right)} + 189 \cos^{2}{\left(x \right)} - 105\right) \cos^{3}{\left(x \right)}}{315}+ \mathrm{constant}$

The answer is:

$\frac{\left(35 \cos^{6}{\left(x \right)} - 135 \cos^{4}{\left(x \right)} + 189 \cos^{2}{\left(x \right)} - 105\right) \cos^{3}{\left(x \right)}}{315}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                  
 |                               7         3         9           5   
 |    7       2             3*cos (x)   cos (x)   cos (x)   3*cos (x)
 | sin (x)*cos (x) dx = C - --------- - ------- + ------- + ---------
 |                              7          3         9          5    
/

$${{35\,\cos ^9x-135\,\cos ^7x+189\,\cos ^5x-105\,\cos ^3x}\over{315 }}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       5/     ___\        7          3          9/     ___\      3/     ___\      9            5            7/     ___\
  3*cos \12*\/ 3 /   3*cos (32)   cos (32)   cos \12*\/ 3 /   cos \12*\/ 3 /   cos (32)   3*cos (32)   3*cos \12*\/ 3 /
- ---------------- - ---------- - -------- - -------------- + -------------- + -------- + ---------- + ----------------
         5               7           3             9                3             9           5               7

$${{35\,\cos ^932-135\,\cos ^732+189\,\cos ^532-105\,\cos ^332}\over{ 315}}-{{35\,\cos ^9\left(4\,3^{{{3}\over{2}}}\right)-135\,\cos ^7 \left(4\,3^{{{3}\over{2}}}\right)+189\,\cos ^5\left(4\,3^{{{3}\over{ 2}}}\right)-105\,\cos ^3\left(4\,3^{{{3}\over{2}}}\right)}\over{315 }}$$

       5/     ___\        7          3          9/     ___\      3/     ___\      9            5            7/     ___\
  3*cos \12*\/ 3 /   3*cos (32)   cos (32)   cos \12*\/ 3 /   cos \12*\/ 3 /   cos (32)   3*cos (32)   3*cos \12*\/ 3 /
- ---------------- - ---------- - -------- - -------------- + -------------- + -------- + ---------- + ----------------
         5               7           3             9                3             9           5               7

$$- \frac{\cos^{3}{\left(32 \right)}}{3} - \frac{3 \cos^{7}{\left(32 \right)}}{7} + \frac{\cos^{3}{\left(12 \sqrt{3} \right)}}{3} + \frac{3 \cos^{7}{\left(12 \sqrt{3} \right)}}{7} - \frac{\cos^{9}{\left(12 \sqrt{3} \right)}}{9} - \frac{3 \cos^{5}{\left(12 \sqrt{3} \right)}}{5} + \frac{\cos^{9}{\left(32 \right)}}{9} + \frac{3 \cos^{5}{\left(32 \right)}}{5}$$

Simplify

Numerical answer [src]

-0.0617950017091718

-0.0617950017091718

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sin^7(x)*cos^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3