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Integral of cos^3x*sin(x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{4}} \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$$
Integral(cos(x)^3*sin(x), (x, pi/4, pi/4))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

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

        1. The integral of is when :

        So, the result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Let .

      Then let and substitute :

      1. Integrate term-by-term:

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

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

          1. The integral of is when :

          So, the result is:

        The result is:

      Now substitute back in:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                               
 |                            4   
 |    3                    cos (x)
 | cos (x)*sin(x) dx = C - -------
 |                            4   
/                                 
$$\int \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx = C - \frac{\cos^{4}{\left(x \right)}}{4}$$
The graph
The answer [src]
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Numerical answer [src]
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    Use the examples entering the upper and lower limits of integration.