Integral of sin(t/2) dt

The solution

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 157         
 ---         
  25         
  /          
 |           
 |     /t\   
 |  sin|-| dt
 |     \2/   
 |           
/            
0

$$\int\limits_{0}^{\frac{157}{25}} \sin{\left(\frac{t}{2} \right)}\, dt$$

Integral(sin(t/2), (t, 0, 157/25))

Detail solution

Let $u = \frac{t}{2}$ .

Then let $du = \frac{dt}{2}$ 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:

$- 2 \cos{\left(\frac{t}{2} \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                        
 |                         
 |    /t\               /t\
 | sin|-| dt = C - 2*cos|-|
 |    \2/               \2/
 |                         
/

$$\int \sin{\left(\frac{t}{2} \right)}\, dt = C - 2 \cos{\left(\frac{t}{2} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /157\
2 - 2*cos|---|
         \ 50/

$$2 - 2 \cos{\left(\frac{157}{50} \right)}$$

         /157\
2 - 2*cos|---|
         \ 50/

$$2 - 2 \cos{\left(\frac{157}{50} \right)}$$

2 - 2*cos(157/50)

Numerical answer [src]

3.99999746345508

3.99999746345508

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

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Integral of sin(t/2) dt

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The solution