Mister Exam

Integral of -sin2x dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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  |  -sin(2*x) dx
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02π(sin(2x))dx\int\limits_{0}^{2 \pi} \left(- \sin{\left(2 x \right)}\right)\, dx
Integral(-sin(2*x), (x, 0, 2*pi))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

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

    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)}

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

  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=C+cos(2x)2\int \left(- \sin{\left(2 x \right)}\right)\, dx = C + \frac{\cos{\left(2 x \right)}}{2}
The graph
0.00.51.01.52.02.53.03.54.04.55.05.56.02-2
The answer [src]
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Numerical answer [src]
5.4435142372695e-22
5.4435142372695e-22

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