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cos^3xsinx
Solving integrals/ cos^3xsinx

Integral of cos^3xsinx dx

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The solution

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∫01sin⁡(x)cos⁡3(x) dx\int\limits_{0}^{1} \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx 
Integral(cos(x)^3*sin(x), (x, 0, 1))
Detail solution

  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:

      ∫u3 du\int u^{3}\, du

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

        ∫(−u3) du=−∫u3 du\int \left(- u^{3}\right)\, 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}

        So, the result is: −u44- \frac{u^{4}}{4}

      Now substitute uu back in:

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

    Method #2

    1. Rewrite the integrand:

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

    2. Let u=sin⁡2(x)u = \sin^{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:

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

      1. Integrate term-by-term:

        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}

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

          ∫(−u2) du=−∫u du2\int \left(- \frac{u}{2}\right)\, 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}

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

      Now substitute uu back in:

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

  2. Add the constant of integration:

    −cos⁡4(x)4+constant- \frac{\cos^{4}{\left(x \right)}}{4}+ \mathrm{constant}

The answer is:

−cos⁡4(x)4+constant- \frac{\cos^{4}{\left(x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
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 |    3                    cos (x)
 | cos (x)*sin(x) dx = C - -------
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−cos⁡4x4-{{\cos ^4x}\over{4}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Plot the graph f(x)
The answer [src] 
       4   
1   cos (1)
- - -------
4      4
14−cos⁡414{{1}\over{4}}-{{\cos ^41}\over{4}} 
=
= 
       4   
1   cos (1)
- - -------
4      4
−cos⁡4(1)4+14- \frac{\cos^{4}{\left(1 \right)}}{4} + \frac{1}{4}
Simplify
Numerical answer [src] 
0.228694717720381
0.228694717720381
The graph
Integral of cos^3xsinx dx

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

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