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Solving integrals/ 0.5*cos(x+pi/4)

Integral of 0.5*cos(x+pi/4) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  4               
  /               
 |                
 |     /    pi\   
 |  cos|x + --|   
 |     \    4 /   
 |  ----------- dx
 |       2        
 |                
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0
∫04cos⁡(x+π4)2 dx\int\limits_{0}^{4} \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx 
Integral(cos(x + pi/4)/2, (x, 0, 4))
Detail solution

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

    ∫cos⁡(x+π4)2 dx=∫cos⁡(x+π4) dx2\int \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx = \frac{\int \cos{\left(x + \frac{\pi}{4} \right)}\, dx}{2}

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

      Then let du=dxdu = dx and substitute dudu:

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

      1. The integral of cosine is sine:

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

      Now substitute uu back in:

      sin⁡(x+π4)\sin{\left(x + \frac{\pi}{4} \right)}

    So, the result is: sin⁡(x+π4)2\frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}

  2. Now simplify:

    sin⁡(x+π4)2\frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                
 |                                 
 |    /    pi\             /    pi\
 | cos|x + --|          sin|x + --|
 |    \    4 /             \    4 /
 | ----------- dx = C + -----------
 |      2                    2     
 |                                 
/
∫cos⁡(x+π4)2 dx=C+sin⁡(x+π4)2\int \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx = C + \frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}
Simplify
The graph
0.04.00.51.01.52.02.53.03.51-1
Plot the graph f(x)
The answer [src] 
   /    pi\        
sin|4 + --|     ___
   \    4 /   \/ 2 
----------- - -----
     2          4
sin⁡(π4+4)2−24\frac{\sin{\left(\frac{\pi}{4} + 4 \right)}}{2} - \frac{\sqrt{2}}{4} 
=
= 
   /    pi\        
sin|4 + --|     ___
   \    4 /   \/ 2 
----------- - -----
     2          4
sin⁡(π4+4)2−24\frac{\sin{\left(\frac{\pi}{4} + 4 \right)}}{2} - \frac{\sqrt{2}}{4} 
sin(4 + pi/4)/2 - sqrt(2)/4
Numerical answer [src] 
-0.852221397214836
-0.852221397214836

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

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