Mister Exam

Other calculators

Integral of cos(2*pi*x)f(t) dy

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                   
  /                   
 |                    
 |  cos(2*pi*x)*f*t dx
 |                    
/                     
0                     
01ftcos(2πx)dx\int\limits_{0}^{1} f t \cos{\left(2 \pi x \right)}\, dx
Integral(cos(2*pi*x)*f*t, (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    ftcos(2πx)dx=ftcos(2πx)dx\int f t \cos{\left(2 \pi x \right)}\, dx = f t \int \cos{\left(2 \pi x \right)}\, dx

    1. Let u=2πxu = 2 \pi x.

      Then let du=2πdxdu = 2 \pi dx and substitute du2π\frac{du}{2 \pi}:

      cos(u)4π2du\int \frac{\cos{\left(u \right)}}{4 \pi^{2}}\, du

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

        cos(u)2πdu=cos(u)du2π\int \frac{\cos{\left(u \right)}}{2 \pi}\, du = \frac{\int \cos{\left(u \right)}\, du}{2 \pi}

        1. The integral of cosine is sine:

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

        So, the result is: sin(u)2π\frac{\sin{\left(u \right)}}{2 \pi}

      Now substitute uu back in:

      sin(2πx)2π\frac{\sin{\left(2 \pi x \right)}}{2 \pi}

    So, the result is: ftsin(2πx)2π\frac{f t \sin{\left(2 \pi x \right)}}{2 \pi}

  2. Add the constant of integration:

    ftsin(2πx)2π+constant\frac{f t \sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}


The answer is:

ftsin(2πx)2π+constant\frac{f t \sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                        
 |                          f*t*sin(2*pi*x)
 | cos(2*pi*x)*f*t dx = C + ---------------
 |                                2*pi     
/                                          
ftsin(2πx)2π{{f\,t\,\sin \left(2\,\pi\,x\right)}\over{2\,\pi}}
The answer [src]
0
fsin(2π)t2π{{f\,\sin \left(2\,\pi\right)\,t}\over{2\,\pi}}
=
=
0
00

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