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Integral of d{x}:
Integral of 1/
Integral of dy/y^2
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Identical expressions
5sin(x/ two +п/ four)
5 sinus of (x divide by 2 plus п divide by 4)
5 sinus of (x divide by two plus п divide by four)
5sinx/2+п/4
5sin(x divide by 2+п divide by 4)
5sin(x/2+п/4)dx
Similar expressions
5sin(x/2-п/4)

Solving integrals/ 5sin(x/2+п/4)

Integral of 5sin(x/2+п/4) dx

The solution

You have entered [src]

 pi                 
  /                 
 |                  
 |       /x   pi\   
 |  5*sin|- + --| dx
 |       \2   4 /   
 |                  
/                   
pi                  
--                  
2

$$\int\limits_{\frac{\pi}{2}}^{\pi} 5 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\, dx$$

Simplify

Detail solution

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

$\int 5 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\, dx = 5 \int \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\, dx$
1. Let $u = \frac{x}{2} + \frac{\pi}{4}$ .
  
  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:
  
  $- 2 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$
So, the result is: $- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$
Now simplify:

$- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$
Add the constant of integration:

$- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}+ \mathrm{constant}$

The answer is:

$- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |      /x   pi\                /pi   x\
 | 5*sin|- + --| dx = C - 10*cos|-- + -|
 |      \2   4 /                \4    2/
 |                                      
/

$$-10\,\cos \left({{x}\over{2}}+{{\pi}\over{4}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___
5*\/ 2

$$5\,\left(2\,\cos \left({{\pi}\over{2}}\right)-2\,\cos \left({{3\, \pi}\over{4}}\right)\right)$$

    ___
5*\/ 2

$$5 \sqrt{2}$$

Simplify

Numerical answer [src]

7.07106781186548

7.07106781186548

The graph

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

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Identical expressions

Similar expressions

Integral of 5sin(x/2+п/4) dx

Limits of integration:

The graph:

Piecewise:

The solution