Integral of (x+3)cosx dx

The solution

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 |  (x + 3)*cos(x) dx
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$$\int\limits_{0}^{\frac{x}{2}} \left(x + 3\right) \cos{\left(x \right)}\, dx$$

Integral((x + 3)*cos(x), (x, 0, x/2))

Detail solution

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

$x \sin{\left(x \right)} + 3 \sin{\left(x \right)} + \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$x \sin{\left(x \right)} + 3 \sin{\left(x \right)} + \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
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 | (x + 3)*cos(x) dx = C + 3*sin(x) + x*sin(x) + cos(x)
 |                                                     
/

$$\int \left(x + 3\right) \cos{\left(x \right)}\, dx = C + x \sin{\left(x \right)} + 3 \sin{\left(x \right)} + \cos{\left(x \right)}$$

Simplify

The answer [src]

                     /x\         
                x*sin|-|         
          /x\        \2/      /x\
-1 + 3*sin|-| + -------- + cos|-|
          \2/      2          \2/

$$\frac{x \sin{\left(\frac{x}{2} \right)}}{2} + 3 \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)} - 1$$

                     /x\         
                x*sin|-|         
          /x\        \2/      /x\
-1 + 3*sin|-| + -------- + cos|-|
          \2/      2          \2/

$$\frac{x \sin{\left(\frac{x}{2} \right)}}{2} + 3 \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)} - 1$$

-1 + 3*sin(x/2) + x*sin(x/2)/2 + cos(x/2)

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

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Identical expressions

Similar expressions

Integral of (x+3)cosx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2