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Identical expressions
((x+ one)cos*x)/ four
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((x plus one) co sinus of e of multiply by x) divide by four
((x+1)cosx)/4
x+1cosx/4
((x+1)cos*x) divide by 4
((x+1)cos*x)/4dx
Similar expressions
((x-1)cos*x)/4

Integral of ((x+1)cos*x)/4 dx

The solution

You have entered [src]

 4*pi                 
   /                  
  |                   
  |  (x + 1)*cos(x)   
  |  -------------- dx
  |        4          
  |                   
 /                    
2*pi

$$\int\limits_{2 \pi}^{4 \pi} \frac{\left(x + 1\right) \cos{\left(x \right)}}{4}\, dx$$

Integral(((x + 1)*cos(x))/4, (x, 2*pi, 4*pi))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\left(x + 1\right) \cos{\left(x \right)}}{4}\, dx = \frac{\int \left(x + 1\right) \cos{\left(x \right)}\, dx}{4}$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\left(x + 1\right) \cos{\left(x \right)} = x \cos{\left(x \right)} + \cos{\left(x \right)}$
  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 $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      
      Now evaluate the sub-integral.
    2. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    The result is: $x \sin{\left(x \right)} + \sin{\left(x \right)} + \cos{\left(x \right)}$
  Method #2
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = x + 1$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    Now evaluate the sub-integral.
  2. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
So, the result is: $\frac{x \sin{\left(x \right)}}{4} + \frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4}$
Now simplify:

$\frac{x \sin{\left(x \right)}}{4} + \frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{4}$
Add the constant of integration:

$\frac{x \sin{\left(x \right)}}{4} + \frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x \sin{\left(x \right)}}{4} + \frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

-1.21665069061698e-15

-1.21665069061698e-15

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

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Identical expressions

Similar expressions

Integral of ((x+1)cos*x)/4 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2