Integral of 2xcos(x/2) dx

The solution

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

Integral((2*x)*cos(x/2), (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = 2 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 2$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $2 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $2 \sin{\left(\frac{x}{2} \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 4 \sin{\left(\frac{x}{2} \right)}\, dx = 4 \int \sin{\left(\frac{x}{2} \right)}\, dx$
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- 2 \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- 2 \cos{\left(\frac{x}{2} \right)}$
So, the result is: $- 8 \cos{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$4 x \sin{\left(\frac{x}{2} \right)} + 8 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$4 x \sin{\left(\frac{x}{2} \right)} + 8 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
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 |        /x\               /x\          /x\
 | 2*x*cos|-| dx = C + 8*cos|-| + 4*x*sin|-|
 |        \2/               \2/          \2/
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$$\int 2 x \cos{\left(\frac{x}{2} \right)}\, dx = C + 4 x \sin{\left(\frac{x}{2} \right)} + 8 \cos{\left(\frac{x}{2} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-8 + 4*sin(1/2) + 8*cos(1/2)

$$-8 + 4 \sin{\left(\frac{1}{2} \right)} + 8 \cos{\left(\frac{1}{2} \right)}$$

-8 + 4*sin(1/2) + 8*cos(1/2)

$$-8 + 4 \sin{\left(\frac{1}{2} \right)} + 8 \cos{\left(\frac{1}{2} \right)}$$

-8 + 4*sin(1/2) + 8*cos(1/2)

Numerical answer [src]

0.938362649539794

0.938362649539794

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

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Integral of 2xcos(x/2) dx

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The solution