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Identical expressions
(one -4x)*cos5x
(1 minus 4x) multiply by co sinus of e of 5x
(one minus 4x) multiply by co sinus of e of 5x
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Similar expressions
(1+4x)*cos5x

Integral of (1-4x)*cos5x dx

The solution

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  1                      
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 |  (1 - 4*x)*cos(5*x) dx
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0

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

Integral((1 - 4*x)*cos(5*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(1 - 4 x\right) \cos{\left(5 x \right)} = - 4 x \cos{\left(5 x \right)} + \cos{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 x \cos{\left(5 x \right)}\right)\, dx = - 4 \int x \cos{\left(5 x \right)}\, dx$
    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 $v{\left(x \right)}$ :
      
      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:
      
      $\int \frac{\sin{\left(5 x \right)}}{5}\, dx = \frac{\int \sin{\left(5 x \right)}\, dx}{5}$
      
      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: $- \frac{\cos{\left(5 x \right)}}{25}$
    So, the result is: $- \frac{4 x \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}$
  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:
    
    $\frac{\sin{\left(5 x \right)}}{5}$
  The result is: $- \frac{4 x \sin{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 1 - 4 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = -4$ .
  
  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:
  
  $\int \left(- \frac{4 \sin{\left(5 x \right)}}{5}\right)\, dx = - \frac{4 \int \sin{\left(5 x \right)}\, dx}{5}$
  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: $\frac{4 \cos{\left(5 x \right)}}{25}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - 4 x\right) \cos{\left(5 x \right)} = - 4 x \cos{\left(5 x \right)} + \cos{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 x \cos{\left(5 x \right)}\right)\, dx = - 4 \int x \cos{\left(5 x \right)}\, dx$
    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 $v{\left(x \right)}$ :
      
      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:
      
      $\int \frac{\sin{\left(5 x \right)}}{5}\, dx = \frac{\int \sin{\left(5 x \right)}\, dx}{5}$
      
      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: $- \frac{\cos{\left(5 x \right)}}{25}$
    So, the result is: $- \frac{4 x \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}$
  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:
    
    $\frac{\sin{\left(5 x \right)}}{5}$
  The result is: $- \frac{4 x \sin{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}$
Add the constant of integration:

$- \frac{4 x \sin{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer is:

$- \frac{4 x \sin{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                
 |                             4*cos(5*x)   sin(5*x)   4*x*sin(5*x)
 | (1 - 4*x)*cos(5*x) dx = C - ---------- + -------- - ------------
 |                                 25          5            5      
/

$$\int \left(1 - 4 x\right) \cos{\left(5 x \right)}\, dx = C - \frac{4 x \sin{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{25}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4    4*cos(5)   3*sin(5)
-- - -------- - --------
25      25         5

$$- \frac{4 \cos{\left(5 \right)}}{25} + \frac{4}{25} - \frac{3 \sin{\left(5 \right)}}{5}$$

4    4*cos(5)   3*sin(5)
-- - -------- - --------
25      25         5

$$- \frac{4 \cos{\left(5 \right)}}{25} + \frac{4}{25} - \frac{3 \sin{\left(5 \right)}}{5}$$

4/25 - 4*cos(5)/25 - 3*sin(5)/5

Numerical answer [src]

0.689968615123767

0.689968615123767

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 (1-4x)*cos5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3