Mister Exam

Other calculators


5sin(x/2+п/4)

Integral of 5sin(x/2+п/4) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 pi                 
  /                 
 |                  
 |       /x   pi\   
 |  5*sin|- + --| dx
 |       \2   4 /   
 |                  
/                   
pi                  
--                  
2                   
π2π5sin(x2+π4)dx\int\limits_{\frac{\pi}{2}}^{\pi} 5 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\, dx
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    5sin(x2+π4)dx=5sin(x2+π4)dx\int 5 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\, dx = 5 \int \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\, dx

    1. Let u=x2+π4u = \frac{x}{2} + \frac{\pi}{4}.

      Then let du=dx2du = \frac{dx}{2} and substitute 2du2 du:

      4sin(u)du\int 4 \sin{\left(u \right)}\, du

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

        2sin(u)du=2sin(u)du\int 2 \sin{\left(u \right)}\, du = 2 \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)}

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

      Now substitute uu back in:

      2cos(x2+π4)- 2 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}

    So, the result is: 10cos(x2+π4)- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}

  2. Now simplify:

    10cos(x2+π4)- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}

  3. Add the constant of integration:

    10cos(x2+π4)+constant- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}+ \mathrm{constant}


The answer is:

10cos(x2+π4)+constant- 10 \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                     
 |                                      
 |      /x   pi\                /pi   x\
 | 5*sin|- + --| dx = C - 10*cos|-- + -|
 |      \2   4 /                \4    2/
 |                                      
/                                       
10cos(x2+π4)-10\,\cos \left({{x}\over{2}}+{{\pi}\over{4}}\right)
The graph
1.61.71.81.92.02.12.22.32.42.52.62.72.82.93.03.1-1010
The answer [src]
    ___
5*\/ 2 
5(2cos(π2)2cos(3π4))5\,\left(2\,\cos \left({{\pi}\over{2}}\right)-2\,\cos \left({{3\, \pi}\over{4}}\right)\right)
=
=
    ___
5*\/ 2 
525 \sqrt{2}
Numerical answer [src]
7.07106781186548
7.07106781186548
The graph
Integral of 5sin(x/2+п/4) dx

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