Integral of sinxsin4x dx

The solution

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$$\int\limits_{0}^{1} \sin{\left(x \right)} \sin{\left(4 x \right)}\, dx$$

Integral(sin(x)*sin(4*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin{\left(x \right)} \sin{\left(4 x \right)} = - 8 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 8 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 8 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{4}\, du$
    1. 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: $- \frac{8 \sin^{5}{\left(x \right)}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 4 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{2}\, du$
    1. 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: $\frac{4 \sin^{3}{\left(x \right)}}{3}$
The result is: $- \frac{8 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3}$
Now simplify:

$\frac{4 \left(5 - 6 \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{15}$
Add the constant of integration:

$\frac{4 \left(5 - 6 \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{15}+ \mathrm{constant}$

The answer is:

$\frac{4 \left(5 - 6 \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              5           3   
 |                          8*sin (x)   4*sin (x)
 | sin(x)*sin(4*x) dx = C - --------- + ---------
 |                              5           3    
/

$$\int \sin{\left(x \right)} \sin{\left(4 x \right)}\, dx = C - \frac{8 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  4*cos(4)*sin(1)   cos(1)*sin(4)
- --------------- + -------------
         15               15

$$\frac{\sin{\left(4 \right)} \cos{\left(1 \right)}}{15} - \frac{4 \sin{\left(1 \right)} \cos{\left(4 \right)}}{15}$$

  4*cos(4)*sin(1)   cos(1)*sin(4)
- --------------- + -------------
         15               15

$$\frac{\sin{\left(4 \right)} \cos{\left(1 \right)}}{15} - \frac{4 \sin{\left(1 \right)} \cos{\left(4 \right)}}{15}$$

-4*cos(4)*sin(1)/15 + cos(1)*sin(4)/15

Numerical answer [src]

0.119412428809625

0.119412428809625

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sinxsin4x dx

Limits of integration:

The graph:

Piecewise:

The solution