Mister Exam

Other calculators

5sin(x/2+п/4)
Solving integrals/ 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
Simplify
Detail solution

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

    ∫5sin⁡(x2+π4) dx=5∫sin⁡(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=2∫sin⁡(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/
 |                                      
/
−10 cos⁡(x2+π4)-10\,\cos \left({{x}\over{2}}+{{\pi}\over{4}}\right)
Simplify
The graph
1.61.71.81.92.02.12.22.32.42.52.62.72.82.93.03.1-1010
Plot the graph f(x)
The answer [src] 
    ___
5*\/ 2
5 (2 cos⁡(π2)−2 cos⁡(3 π4))5\,\left(2\,\cos \left({{\pi}\over{2}}\right)-2\,\cos \left({{3\, \pi}\over{4}}\right)\right) 
=
= 
    ___
5*\/ 2
525 \sqrt{2}
Simplify
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.

    © Mister Exam – Calculator