Integral of 4sin^3*x*cosx dx

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

   $$\int 4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = 4 \int \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

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

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

         $$\int u^{3}\, du$$

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

            $$\int u^{3}\, du = \frac{u^{4}}{4}$$

         Now substitute $u$ back in:

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

      Method #2

      1. Rewrite the integrand:

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

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

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

         $$\int \left(\frac{u}{2} + \frac{1}{2}\right)\, du$$

         1. Integrate term-by-term:

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

               $$\int \frac{u}{2}\, du = \frac{\int u\, du}{2}$$

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

                  $$\int u\, du = \frac{u^{2}}{2}$$

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

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

               $$\int \frac{1}{2}\, du = \frac{u}{2}$$

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

         Now substitute $u$ back in:

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

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

2. Add the constant of integration:

   $$\sin^{4}{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\sin^{4}{\left(x \right)}+ \mathrm{constant}$$