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Identical expressions
sin(2x)*sin(5x)
sinus of (2x) multiply by sinus of (5x)
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sin(2x)*sin(5x)dx
Similar expressions
sin2x*sin5x

Integral of sin(2x)*sin(5x) dx

The solution

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 pi                     
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 |  sin(2*x)*sin(5*x) dx
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0

$$\int\limits_{0}^{\frac{\pi}{2}} \sin{\left(2 x \right)} \sin{\left(5 x \right)}\, dx$$

Integral(sin(2*x)*sin(5*x), (x, 0, pi/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 \sin{\left(x \right)} \sin{\left(5 x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \sin{\left(5 x \right)} \cos{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\sin{\left(x \right)} \sin{\left(5 x \right)} \cos{\left(x \right)} = 16 \sin^{6}{\left(x \right)} \cos{\left(x \right)} - 20 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 5 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 16 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = 16 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      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: $\frac{16 \sin^{7}{\left(x \right)}}{7}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 20 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 20 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      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: $- 4 \sin^{5}{\left(x \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 5 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 5 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      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: $\frac{5 \sin^{3}{\left(x \right)}}{3}$
    The result is: $\frac{16 \sin^{7}{\left(x \right)}}{7} - 4 \sin^{5}{\left(x \right)} + \frac{5 \sin^{3}{\left(x \right)}}{3}$
  So, the result is: $\frac{32 \sin^{7}{\left(x \right)}}{7} - 8 \sin^{5}{\left(x \right)} + \frac{10 \sin^{3}{\left(x \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\sin{\left(2 x \right)} \sin{\left(5 x \right)} = 32 \sin^{6}{\left(x \right)} \cos{\left(x \right)} - 40 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 10 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 32 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = 32 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $\frac{32 \sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 40 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 40 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $- 8 \sin^{5}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 10 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 10 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    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: $\frac{10 \sin^{3}{\left(x \right)}}{3}$
  The result is: $\frac{32 \sin^{7}{\left(x \right)}}{7} - 8 \sin^{5}{\left(x \right)} + \frac{10 \sin^{3}{\left(x \right)}}{3}$
Now simplify:

$\frac{2 \left(48 \sin^{4}{\left(x \right)} - 84 \sin^{2}{\left(x \right)} + 35\right) \sin^{3}{\left(x \right)}}{21}$
Add the constant of integration:

$\frac{2 \left(48 \sin^{4}{\left(x \right)} - 84 \sin^{2}{\left(x \right)} + 35\right) \sin^{3}{\left(x \right)}}{21}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(48 \sin^{4}{\left(x \right)} - 84 \sin^{2}{\left(x \right)} + 35\right) \sin^{3}{\left(x \right)}}{21}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             3            7   
 |                                 5      10*sin (x)   32*sin (x)
 | sin(2*x)*sin(5*x) dx = C - 8*sin (x) + ---------- + ----------
 |                                            3            7     
/

$$\int \sin{\left(2 x \right)} \sin{\left(5 x \right)}\, dx = C + \frac{32 \sin^{7}{\left(x \right)}}{7} - 8 \sin^{5}{\left(x \right)} + \frac{10 \sin^{3}{\left(x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2/21

$$- \frac{2}{21}$$

-2/21

$$- \frac{2}{21}$$

-2/21

Numerical answer [src]

-0.0952380952380952

-0.0952380952380952

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

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Identical expressions

Similar expressions

Integral of sin(2x)*sin(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2