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Identical expressions
(7x+ two)*cos5x
(7x plus 2) multiply by co sinus of e of 5x
(7x plus two) multiply by co sinus of e of 5x
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7x+2cos5x
(7x+2)*cos5xdx
Similar expressions
(7x-2)*cos5x

Integral of (7x+2)*cos5x dx

The solution

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 |  (7*x + 2)*cos(5*x) dx
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$$\int\limits_{0}^{1} \left(7 x + 2\right) \cos{\left(5 x \right)}\, dx$$

Integral((7*x + 2)*cos(5*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(7 x + 2\right) \cos{\left(5 x \right)} = 7 x \cos{\left(5 x \right)} + 2 \cos{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 7 x \cos{\left(5 x \right)}\, dx = 7 \int x \cos{\left(5 x \right)}\, dx$
    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: $\frac{7 x \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \cos{\left(5 x \right)}\, dx = 2 \int \cos{\left(5 x \right)}\, dx$
    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: $\frac{7 x \sin{\left(5 x \right)}}{5} + \frac{2 \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 7 x + 2$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 7$ .
  
  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:
  
  $\int \frac{7 \sin{\left(5 x \right)}}{5}\, dx = \frac{7 \int \sin{\left(5 x \right)}\, dx}{5}$
  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: $- \frac{7 \cos{\left(5 x \right)}}{25}$
Method #3
1. Rewrite the integrand:
  
  $\left(7 x + 2\right) \cos{\left(5 x \right)} = 7 x \cos{\left(5 x \right)} + 2 \cos{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 7 x \cos{\left(5 x \right)}\, dx = 7 \int x \cos{\left(5 x \right)}\, dx$
    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: $\frac{7 x \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \cos{\left(5 x \right)}\, dx = 2 \int \cos{\left(5 x \right)}\, dx$
    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: $\frac{7 x \sin{\left(5 x \right)}}{5} + \frac{2 \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}$
Add the constant of integration:

$\frac{7 x \sin{\left(5 x \right)}}{5} + \frac{2 \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer is:

$\frac{7 x \sin{\left(5 x \right)}}{5} + \frac{2 \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                  
 |                             2*sin(5*x)   7*cos(5*x)   7*x*sin(5*x)
 | (7*x + 2)*cos(5*x) dx = C + ---------- + ---------- + ------------
 |                                 5            25            5      
/

$$\int \left(7 x + 2\right) \cos{\left(5 x \right)}\, dx = C + \frac{7 x \sin{\left(5 x \right)}}{5} + \frac{2 \sin{\left(5 x \right)}}{5} + \frac{7 \cos{\left(5 x \right)}}{25}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  7    7*cos(5)   9*sin(5)
- -- + -------- + --------
  25      25         5

$$\frac{9 \sin{\left(5 \right)}}{5} - \frac{7}{25} + \frac{7 \cos{\left(5 \right)}}{25}$$

  7    7*cos(5)   9*sin(5)
- -- + -------- + --------
  25      25         5

$$\frac{9 \sin{\left(5 \right)}}{5} - \frac{7}{25} + \frac{7 \cos{\left(5 \right)}}{25}$$

-7/25 + 7*cos(5)/25 + 9*sin(5)/5

Numerical answer [src]

-1.92663828246395

-1.92663828246395

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 (7x+2)*cos5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3