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Identical expressions
four *sin(5x)^ three
4 multiply by sinus of (5x) cubed
four multiply by sinus of (5x) to the power of three
4*sin(5x)3
4*sin5x3
4*sin(5x)³
4*sin(5x) to the power of 3
4sin(5x)^3
4sin(5x)3
4sin5x3
4sin5x^3
4*sin(5x)^3dx

Solving integrals/ 4*sin(5x)^3

Integral of 4*sin(5x)^3 dx

The solution

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  1               
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 |       3        
 |  4*sin (5*x) dx
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0

$$\int\limits_{0}^{1} 4 \sin^{3}{\left(5 x \right)}\, dx$$

Integral(4*sin(5*x)^3, (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 4 \sin^{3}{\left(5 x \right)}\, dx = 4 \int \sin^{3}{\left(5 x \right)}\, dx$
1. Rewrite the integrand:
  
  $\sin^{3}{\left(5 x \right)} = \left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)}$
2. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \cos{\left(5 x \right)}$ .
    
    Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $du$ :
    
    $\int \left(\frac{u^{2}}{5} - \frac{1}{5}\right)\, du$
    1. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{2}}{5}\, du = \frac{\int u^{2}\, du}{5}$
      
      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: $\frac{u^{3}}{15}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \left(- \frac{1}{5}\right)\, du = - \frac{u}{5}$
      
      The result is: $\frac{u^{3}}{15} - \frac{u}{5}$
    Now substitute $u$ back in:
    
    $\frac{\cos^{3}{\left(5 x \right)}}{15} - \frac{\cos{\left(5 x \right)}}{5}$
  Method #2
  1. Rewrite the integrand:
    
    $\left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)} = - \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)} + \sin{\left(5 x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\right)\, dx = - \int \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\, dx$
      
      Let $u = \cos{\left(5 x \right)}$ .
      
      Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
      
      $\int \frac{u^{2}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{2}}{5}\right)\, du = - \frac{\int u^{2}\, du}{5}$
      
      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: $- \frac{u^{3}}{15}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(5 x \right)}}{15}$
      
      So, the result is: $\frac{\cos^{3}{\left(5 x \right)}}{15}$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
    The result is: $\frac{\cos^{3}{\left(5 x \right)}}{15} - \frac{\cos{\left(5 x \right)}}{5}$
  Method #3
  1. Rewrite the integrand:
    
    $\left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)} = - \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)} + \sin{\left(5 x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\right)\, dx = - \int \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\, dx$
      
      Let $u = \cos{\left(5 x \right)}$ .
      
      Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
      
      $\int \frac{u^{2}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{2}}{5}\right)\, du = - \frac{\int u^{2}\, du}{5}$
      
      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: $- \frac{u^{3}}{15}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(5 x \right)}}{15}$
      
      So, the result is: $\frac{\cos^{3}{\left(5 x \right)}}{15}$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
    The result is: $\frac{\cos^{3}{\left(5 x \right)}}{15} - \frac{\cos{\left(5 x \right)}}{5}$
So, the result is: $\frac{4 \cos^{3}{\left(5 x \right)}}{15} - \frac{4 \cos{\left(5 x \right)}}{5}$
Now simplify:

$- \frac{3 \cos{\left(5 x \right)}}{5} + \frac{\cos{\left(15 x \right)}}{15}$
Add the constant of integration:

$- \frac{3 \cos{\left(5 x \right)}}{5} + \frac{\cos{\left(15 x \right)}}{15}+ \mathrm{constant}$

The answer is:

$- \frac{3 \cos{\left(5 x \right)}}{5} + \frac{\cos{\left(15 x \right)}}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                        3     
 |      3               4*cos(5*x)   4*cos (5*x)
 | 4*sin (5*x) dx = C - ---------- + -----------
 |                          5             15    
/

$$\int 4 \sin^{3}{\left(5 x \right)}\, dx = C + \frac{4 \cos^{3}{\left(5 x \right)}}{15} - \frac{4 \cos{\left(5 x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                     3   
8    4*cos(5)   4*cos (5)
-- - -------- + ---------
15      5           15

$$- \frac{4 \cos{\left(5 \right)}}{5} + \frac{4 \cos^{3}{\left(5 \right)}}{15} + \frac{8}{15}$$

                     3   
8    4*cos(5)   4*cos (5)
-- - -------- + ---------
15      5           15

$$- \frac{4 \cos{\left(5 \right)}}{5} + \frac{4 \cos^{3}{\left(5 \right)}}{15} + \frac{8}{15}$$

Упростить

Numerical answer [src]

0.312490161198143

0.312490161198143

The graph

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

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Identical expressions

Integral of 4*sin(5x)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3