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Solving integrals/ sin(2*x+pi/4)

Integral of sin(2*x+pi/4) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫0π2sin⁡(2x+π4) dx\int\limits_{0}^{\frac{\pi}{2}} \sin{\left(2 x + \frac{\pi}{4} \right)}\, dx 
Integral(sin(2*x + pi/4), (x, 0, pi/2))
Detail solution

  1. Let u=2x+π4u = 2 x + \frac{\pi}{4}.

    Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

    ∫sin⁡(u)2 du\int \frac{\sin{\left(u \right)}}{2}\, du

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

      ∫sin⁡(u) du=∫sin⁡(u) du2\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}

      1. The integral of sine is negative cosine:

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

      So, the result is: −cos⁡(u)2- \frac{\cos{\left(u \right)}}{2}

    Now substitute uu back in:

    −cos⁡(2x+π4)2- \frac{\cos{\left(2 x + \frac{\pi}{4} \right)}}{2}

  2. Now simplify:

    −cos⁡(2x+π4)2- \frac{\cos{\left(2 x + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

    −cos⁡(2x+π4)2+constant- \frac{\cos{\left(2 x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer is:

−cos⁡(2x+π4)2+constant- \frac{\cos{\left(2 x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                          /      pi\
 |                        cos|2*x + --|
 |    /      pi\             \      4 /
 | sin|2*x + --| dx = C - -------------
 |    \      4 /                2      
 |                                     
/
∫sin⁡(2x+π4) dx=C−cos⁡(2x+π4)2\int \sin{\left(2 x + \frac{\pi}{4} \right)}\, dx = C - \frac{\cos{\left(2 x + \frac{\pi}{4} \right)}}{2}
Simplify
The graph
0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.52-2
Plot the graph f(x)
The answer [src] 
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\/ 2 
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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] 
0.707106781186548
0.707106781186548

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

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