Integral of cos5xsinxdx dx

The solution

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

Integral(cos(5*x)*sin(x)*1, (x, 0, 0))

Detail solution

Rewrite the integrand:

$\cos{\left(5 x \right)} \sin{\left(x \right)} 1 = 16 \sin{\left(x \right)} \cos^{5}{\left(x \right)} - 20 \sin{\left(x \right)} \cos^{3}{\left(x \right)} + 5 \sin{\left(x \right)} \cos{\left(x \right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 16 \sin{\left(x \right)} \cos^{5}{\left(x \right)}\, dx = 16 \int \sin{\left(x \right)} \cos^{5}{\left(x \right)}\, dx$
  1. 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 $- du$ :
      
      $\int u^{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{5}\right)\, du = - \int u^{5}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{5}\, du = \frac{u^{6}}{6}$
      
      So, the result is: $- \frac{u^{6}}{6}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{6}{\left(x \right)}}{6}$
    Method #2
    1. Rewrite the integrand:
      
      $\sin{\left(x \right)} \cos^{5}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}$
    2. Let $u = 1 - \sin^{2}{\left(x \right)}$ .
      
      Then let $du = - 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
      
      $\int \frac{u^{2}}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{2}}{2}\right)\, du = - \frac{\int u^{2}\, du}{2}$
      
      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: $- \frac{u^{3}}{6}$
      
      Now substitute $u$ back in:
      
      $- \frac{\left(1 - \sin^{2}{\left(x \right)}\right)^{3}}{6}$
  So, the result is: $- \frac{8 \cos^{6}{\left(x \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 20 \sin{\left(x \right)} \cos^{3}{\left(x \right)}\right)\, dx = - 20 \int \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int u^{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{3}\right)\, du = - \int u^{3}\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{3}\, du = \frac{u^{4}}{4}$
      So, the result is: $- \frac{u^{4}}{4}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{4}{\left(x \right)}}{4}$
  So, the result is: $5 \cos^{4}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 5 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 5 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int u\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u\right)\, du = - \int u\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u\, du = \frac{u^{2}}{2}$
      So, the result is: $- \frac{u^{2}}{2}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(x \right)}}{2}$
  So, the result is: $- \frac{5 \cos^{2}{\left(x \right)}}{2}$
The result is: $- \frac{8 \cos^{6}{\left(x \right)}}{3} + 5 \cos^{4}{\left(x \right)} - \frac{5 \cos^{2}{\left(x \right)}}{2}$
Now simplify:

$\frac{\left(- 16 \cos^{4}{\left(x \right)} + 30 \cos^{2}{\left(x \right)} - 15\right) \cos^{2}{\left(x \right)}}{6}$
Add the constant of integration:

$\frac{\left(- 16 \cos^{4}{\left(x \right)} + 30 \cos^{2}{\left(x \right)} - 15\right) \cos^{2}{\left(x \right)}}{6}+ \mathrm{constant}$

The answer is:

$\frac{\left(- 16 \cos^{4}{\left(x \right)} + 30 \cos^{2}{\left(x \right)} - 15\right) \cos^{2}{\left(x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            6           2   
 |                                 4      8*cos (x)   5*cos (x)
 | cos(5*x)*sin(x)*1 dx = C + 5*cos (x) - --------- - ---------
 |                                            3           2    
/

$$\int \cos{\left(5 x \right)} \sin{\left(x \right)} 1\, dx = C - \frac{8 \cos^{6}{\left(x \right)}}{3} + 5 \cos^{4}{\left(x \right)} - \frac{5 \cos^{2}{\left(x \right)}}{2}$$

Simplify

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Numerical answer [src]

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The graph

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

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Identical expressions

Integral of cos5xsinxdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2