Mister Exam

Integral of sin(2*x) dx

Limits of integration:

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The solution

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01sin(2x)dx\int\limits_{0}^{1} \sin{\left(2 x \right)}\, dx
Integral(sin(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2xu = 2 x.

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

      sin(u)2du\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)2- \frac{\cos{\left(2 x \right)}}{2}

    Method #2

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

      2sin(x)cos(x)dx=2sin(x)cos(x)dx\int 2 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx

      1. There are multiple ways to do this integral.

        Method #1

        1. Let u=cos(x)u = \cos{\left(x \right)}.

          Then let du=sin(x)dxdu = - \sin{\left(x \right)} dx and substitute du- du:

          (u)du\int \left(- u\right)\, du

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

            udu=udu\int u\, du = - \int u\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

              udu=u22\int u\, du = \frac{u^{2}}{2}

            So, the result is: u22- \frac{u^{2}}{2}

          Now substitute uu back in:

          cos2(x)2- \frac{\cos^{2}{\left(x \right)}}{2}

        Method #2

        1. Let u=sin(x)u = \sin{\left(x \right)}.

          Then let du=cos(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          udu\int u\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

            udu=u22\int u\, du = \frac{u^{2}}{2}

          Now substitute uu back in:

          sin2(x)2\frac{\sin^{2}{\left(x \right)}}{2}

      So, the result is: cos2(x)- \cos^{2}{\left(x \right)}

  2. Add the constant of integration:

    cos(2x)2+constant- \frac{\cos{\left(2 x \right)}}{2}+ \mathrm{constant}


The answer is:

cos(2x)2+constant- \frac{\cos{\left(2 x \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                          
 |                   cos(2*x)
 | sin(2*x) dx = C - --------
 |                      2    
/                            
sin(2x)dx=Ccos(2x)2\int \sin{\left(2 x \right)}\, dx = C - \frac{\cos{\left(2 x \right)}}{2}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
The answer [src]
1   cos(2)
- - ------
2     2   
12cos(2)2\frac{1}{2} - \frac{\cos{\left(2 \right)}}{2}
=
=
1   cos(2)
- - ------
2     2   
12cos(2)2\frac{1}{2} - \frac{\cos{\left(2 \right)}}{2}
1/2 - cos(2)/2
Numerical answer [src]
0.708073418273571
0.708073418273571
The graph
Integral of sin(2*x) dx

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