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Solving integrals/ sin(x+(pi/3))

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

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
 2*p              
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 |  sin|x + --| dx
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∫02psin⁡(x+π3) dx\int\limits_{0}^{2 p} \sin{\left(x + \frac{\pi}{3} \right)}\, dx 
Integral(sin(x + pi/3), (x, 0, 2*p))
Detail solution

  1. Let u=x+π3u = x + \frac{\pi}{3}.

    Then let du=dxdu = dx and substitute dudu:

    ∫sin⁡(u) du\int \sin{\left(u \right)}\, du

    1. The integral of sine is negative cosine:

      ∫sin⁡(u) du=−cos⁡(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

    Now substitute uu back in:

    −cos⁡(x+π3)- \cos{\left(x + \frac{\pi}{3} \right)}

  2. Now simplify:

    −cos⁡(x+π3)- \cos{\left(x + \frac{\pi}{3} \right)}

  3. Add the constant of integration:

    −cos⁡(x+π3)+constant- \cos{\left(x + \frac{\pi}{3} \right)}+ \mathrm{constant}

The answer is:

−cos⁡(x+π3)+constant- \cos{\left(x + \frac{\pi}{3} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
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 |    /    pi\             /    pi\
 | sin|x + --| dx = C - cos|x + --|
 |    \    3 /             \    3 /
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∫sin⁡(x+π3) dx=C−cos⁡(x+π3)\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 /
12−cos⁡(2p+π3)\frac{1}{2} - \cos{\left(2 p + \frac{\pi}{3} \right)} 
=
= 
1      /      pi\
- - cos|2*p + --|
2      \      3 /
12−cos⁡(2p+π3)\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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