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2sin^3x*cosx
Solving integrals/ 2sin^3x*cosx

Integral of 2sin^3x*cosx dx

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∫012sin⁡3(x)cos⁡(x) dx\int\limits_{0}^{1} 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx 
Integral(2*sin(x)^3*cos(x), (x, 0, 1))
Detail solution

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

    ∫2sin⁡3(x)cos⁡(x) dx=2∫sin⁡3(x)cos⁡(x) dx\int 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx

    1. There are multiple ways to do this integral.

      Method #1

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

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

        ∫u3 du\int u^{3}\, du

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

          ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

        Now substitute uu back in:

        sin⁡4(x)4\frac{\sin^{4}{\left(x \right)}}{4}

      Method #2

      1. Rewrite the integrand:

        sin⁡3(x)cos⁡(x)=(1−cos⁡2(x))sin⁡(x)cos⁡(x)\sin^{3}{\left(x \right)} \cos{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}

      2. Let u=−cos⁡2(x)u = - \cos^{2}{\left(x \right)}.

        Then let du=2sin⁡(x)cos⁡(x)dxdu = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx and substitute dudu:

        ∫(u2+12) du\int \left(\frac{u}{2} + \frac{1}{2}\right)\, du

        1. Integrate term-by-term:

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

            ∫u2 du=∫u du2\int \frac{u}{2}\, du = \frac{\int u\, du}{2}

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

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

            So, the result is: u24\frac{u^{2}}{4}

          1. The integral of a constant is the constant times the variable of integration:

            ∫12 du=u2\int \frac{1}{2}\, du = \frac{u}{2}

          The result is: u24+u2\frac{u^{2}}{4} + \frac{u}{2}

        Now substitute uu back in:

        cos⁡4(x)4−cos⁡2(x)2\frac{\cos^{4}{\left(x \right)}}{4} - \frac{\cos^{2}{\left(x \right)}}{2}

    So, the result is: sin⁡4(x)2\frac{\sin^{4}{\left(x \right)}}{2}

  2. Add the constant of integration:

    sin⁡4(x)2+constant\frac{\sin^{4}{\left(x \right)}}{2}+ \mathrm{constant}

The answer is:

sin⁡4(x)2+constant\frac{\sin^{4}{\left(x \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
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 |      3                    sin (x)
 | 2*sin (x)*cos(x) dx = C + -------
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∫2sin⁡3(x)cos⁡(x) dx=C+sin⁡4(x)2\int 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{4}{\left(x \right)}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
   4   
sin (1)
-------
   2
sin⁡4(1)2\frac{\sin^{4}{\left(1 \right)}}{2} 
=
= 
   4   
sin (1)
-------
   2
sin⁡4(1)2\frac{\sin^{4}{\left(1 \right)}}{2}
Упростить
Numerical answer [src] 
0.25068398283281
0.25068398283281
The graph
Integral of 2sin^3x*cosx dx

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

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