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Identical expressions
four sin(x+(pi/4))
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Integral of 4sin(x+(pi/4)) dx

The solution

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  1                 
  /                 
 |                  
 |       /    pi\   
 |  4*sin|x + --| dx
 |       \    4 /   
 |                  
/                   
0

$$\int\limits_{0}^{1} 4 \sin{\left(x + \frac{\pi}{4} \right)}\, dx$$

Integral(4*sin(x + pi/4), (x, 0, 1))

Detail solution

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

$\int 4 \sin{\left(x + \frac{\pi}{4} \right)}\, dx = 4 \int \sin{\left(x + \frac{\pi}{4} \right)}\, dx$
1. Let $u = x + \frac{\pi}{4}$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \sin{\left(u \right)}\, du$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- \cos{\left(x + \frac{\pi}{4} \right)}$
So, the result is: $- 4 \cos{\left(x + \frac{\pi}{4} \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int 4 \sin{\left(x + \frac{\pi}{4} \right)}\, dx = C - 4 \cos{\left(x + \frac{\pi}{4} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       /    pi\       ___
- 4*cos|1 + --| + 2*\/ 2 
       \    4 /

$$- 4 \cos{\left(\frac{\pi}{4} + 1 \right)} + 2 \sqrt{2}$$

       /    pi\       ___
- 4*cos|1 + --| + 2*\/ 2 
       \    4 /

$$- 4 \cos{\left(\frac{\pi}{4} + 1 \right)} + 2 \sqrt{2}$$

-4*cos(1 + pi/4) + 2*sqrt(2)

Numerical answer [src]

3.68026078538337

3.68026078538337

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

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Integral of 4sin(x+(pi/4)) dx

Limits of integration:

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Piecewise:

The solution