Mister Exam

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(sin(five *x))^ three
( sinus of (5 multiply by x)) cubed
( sinus of (five multiply by x)) to the power of three
(sin(5*x))3
sin5*x3
(sin(5*x))³
(sin(5*x)) to the power of 3
(sin(5x))^3
(sin(5x))3
sin5x3
sin5x^3
(sin(5*x))^3dx
Similar expressions
cos(5*x)*dx/sin(5*x)^(3/2)
4*sin(5x)^3

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

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

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\sin^{3}{\left(5 x \right)} = \left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)}$
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:
    1. 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: $\frac{u^{3}}{15}$
    1. 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:
    
    $\int \left(- \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\right)\, dx = - \int \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\, dx$
    1. 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 $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{25}\, du$
    1. 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:
    
    $\int \left(- \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\right)\, dx = - \int \sin{\left(5 x \right)} \cos^{2}{\left(5 x \right)}\, dx$
    1. 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 $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{25}\, du$
    1. 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: $\frac{\cos^{3}{\left(5 x \right)}}{15} - \frac{\cos{\left(5 x \right)}}{5}$
Now simplify:

$- \frac{3 \cos{\left(5 x \right)}}{20} + \frac{\cos{\left(15 x \right)}}{60}$
Add the constant of integration:

$- \frac{3 \cos{\left(5 x \right)}}{20} + \frac{\cos{\left(15 x \right)}}{60}+ \mathrm{constant}$

The answer is:

$- \frac{3 \cos{\left(5 x \right)}}{20} + \frac{\cos{\left(15 x \right)}}{60}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$${{{{\cos ^3\left(5\,x\right)}\over{3}}-\cos \left(5\,x\right) }\over{5}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                 3   
2    cos(5)   cos (5)
-- - ------ + -------
15     5         15

$${{\cos ^35-3\,\cos 5}\over{15}}+{{2}\over{15}}$$

                 3   
2    cos(5)   cos (5)
-- - ------ + -------
15     5         15

$$- \frac{\cos{\left(5 \right)}}{5} + \frac{\cos^{3}{\left(5 \right)}}{15} + \frac{2}{15}$$

Упростить

Numerical answer [src]

0.0781225402995357

0.0781225402995357

The graph

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

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How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3