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cos(5x)*(x- two)
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Similar expressions
cos(5x)*(x+2)

Integral of cos(5x)*(x-2) dx

The solution

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  1                    
  /                    
 |                     
 |  cos(5*x)*(x - 2) dx
 |                     
/                      
0

$$\int\limits_{0}^{1} \left(x - 2\right) \cos{\left(5 x \right)}\, dx$$

Integral(cos(5*x)*(x - 2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(x - 2\right) \cos{\left(5 x \right)} = x \cos{\left(5 x \right)} - 2 \cos{\left(5 x \right)}$
2. Integrate term-by-term:
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\cos{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(5 x \right)}}{5}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(5 x \right)}}{5}\, dx = \frac{\int \sin{\left(5 x \right)}\, dx}{5}$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      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:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
    So, the result is: $- \frac{\cos{\left(5 x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cos{\left(5 x \right)}\right)\, dx = - 2 \int \cos{\left(5 x \right)}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\cos{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(5 x \right)}}{5}$
    So, the result is: $- \frac{2 \sin{\left(5 x \right)}}{5}$
  The result is: $\frac{x \sin{\left(5 x \right)}}{5} - \frac{2 \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = x - 2$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\cos{\left(u \right)}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(5 x \right)}}{5}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(5 x \right)}}{5}\, dx = \frac{\int \sin{\left(5 x \right)}\, dx}{5}$
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      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:
    
    $- \frac{\cos{\left(5 x \right)}}{5}$
  So, the result is: $- \frac{\cos{\left(5 x \right)}}{25}$
Method #3
1. Rewrite the integrand:
  
  $\left(x - 2\right) \cos{\left(5 x \right)} = x \cos{\left(5 x \right)} - 2 \cos{\left(5 x \right)}$
2. Integrate term-by-term:
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\cos{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(5 x \right)}}{5}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(5 x \right)}}{5}\, dx = \frac{\int \sin{\left(5 x \right)}\, dx}{5}$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      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:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
    So, the result is: $- \frac{\cos{\left(5 x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cos{\left(5 x \right)}\right)\, dx = - 2 \int \cos{\left(5 x \right)}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\cos{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(5 x \right)}}{5}$
    So, the result is: $- \frac{2 \sin{\left(5 x \right)}}{5}$
  The result is: $\frac{x \sin{\left(5 x \right)}}{5} - \frac{2 \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}$
Add the constant of integration:

$\frac{x \sin{\left(5 x \right)}}{5} - \frac{2 \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer is:

$\frac{x \sin{\left(5 x \right)}}{5} - \frac{2 \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                           2*sin(5*x)   cos(5*x)   x*sin(5*x)
 | cos(5*x)*(x - 2) dx = C - ---------- + -------- + ----------
 |                               5           25          5     
/

$$\int \left(x - 2\right) \cos{\left(5 x \right)}\, dx = C + \frac{x \sin{\left(5 x \right)}}{5} - \frac{2 \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1    sin(5)   cos(5)
- -- - ------ + ------
  25     5        25

$$- \frac{1}{25} + \frac{\cos{\left(5 \right)}}{25} - \frac{\sin{\left(5 \right)}}{5}$$

  1    sin(5)   cos(5)
- -- - ------ + ------
  25     5        25

$$- \frac{1}{25} + \frac{\cos{\left(5 \right)}}{25} - \frac{\sin{\left(5 \right)}}{5}$$

-1/25 - sin(5)/5 + cos(5)/25

Numerical answer [src]

0.163131342351157

0.163131342351157

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

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Identical expressions

Similar expressions

Integral of cos(5x)*(x-2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3