Integral of sin(x+(pi/3)) dx

The solution

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 2*p              
  /               
 |                
 |     /    pi\   
 |  sin|x + --| dx
 |     \    3 /   
 |                
/                 
0

$$\int\limits_{0}^{2 p} \sin{\left(x + \frac{\pi}{3} \right)}\, dx$$

Integral(sin(x + pi/3), (x, 0, 2*p))

Detail solution

Let $u = x + \frac{\pi}{3}$ .

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:

$- \cos{\left(x + \frac{\pi}{3} \right)}$
Now simplify:

$- \cos{\left(x + \frac{\pi}{3} \right)}$
Add the constant of integration:

$- \cos{\left(x + \frac{\pi}{3} \right)}+ \mathrm{constant}$

The answer is:

$- \cos{\left(x + \frac{\pi}{3} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                 
 |    /    pi\             /    pi\
 | sin|x + --| dx = C - cos|x + --|
 |    \    3 /             \    3 /
 |                                 
/

$$\int \sin{\left(x + \frac{\pi}{3} \right)}\, dx = C - \cos{\left(x + \frac{\pi}{3} \right)}$$

Simplify

The answer [src]

1      /      pi\
- - cos|2*p + --|
2      \      3 /

$$\frac{1}{2} - \cos{\left(2 p + \frac{\pi}{3} \right)}$$

1      /      pi\
- - cos|2*p + --|
2      \      3 /

$$\frac{1}{2} - \cos{\left(2 p + \frac{\pi}{3} \right)}$$

1/2 - cos(2*p + pi/3)

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

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Integral of sin(x+(pi/3)) dx

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