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Solving integrals/ sin(x+pi/4)

Integral of sin(x+pi/4) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
 pi               
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∫0π4sin⁡(x+π4) dx\int\limits_{0}^{\frac{\pi}{4}} \sin{\left(x + \frac{\pi}{4} \right)}\, dx 
Integral(sin(x + pi/4), (x, 0, pi/4))
Detail solution

  1. Let u=x+π4u = x + \frac{\pi}{4}.

    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+π4)- \cos{\left(x + \frac{\pi}{4} \right)}

  2. Now simplify:

    −cos⁡(x+π4)- \cos{\left(x + \frac{\pi}{4} \right)}

  3. Add the constant of integration:

    −cos⁡(x+π4)+constant- \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}

The answer is:

−cos⁡(x+π4)+constant- \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
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 |    /    pi\             /    pi\
 | sin|x + --| dx = C - cos|x + --|
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∫sin⁡(x+π4) dx=C−cos⁡(x+π4)\int \sin{\left(x + \frac{\pi}{4} \right)}\, dx = C - \cos{\left(x + \frac{\pi}{4} \right)}
Simplify
The graph
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.752-2
Plot the graph f(x)
The answer [src] 
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  2
22\frac{\sqrt{2}}{2} 
=
= 
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  2
22\frac{\sqrt{2}}{2} 
sqrt(2)/2
Numerical answer [src] 
0.707106781186547
0.707106781186547

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

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