Integral of 4sin(x/2) dx

The solution

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$$\int\limits_{0}^{\pi} 4 \sin{\left(\frac{x}{2} \right)}\, dx$$

Integral(4*sin(x/2), (x, 0, pi))

Detail solution

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 $2 du$ :
  
  $\int 4 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
    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: $- 8 \cos{\left(\frac{x}{2} \right)}$
Now simplify:

$- 8 \cos{\left(\frac{x}{2} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

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The answer [src]

$$8$$

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Numerical answer [src]

8.0

8.0

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

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Integral of 4sin(x/2) dx

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