Integral of cos^3x dx

1. Rewrite the integrand:

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

2. 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 $du$:

      $$\int \left(1 - u^{2}\right)\, du$$

      1. Integrate term-by-term:

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

            $$\int 1\, du = u$$

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

            $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

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

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         The result is: $- \frac{u^{3}}{3} + u$

      Now substitute $u$ back in:

      $$- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$

   Method #2

   1. Rewrite the integrand:

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

   2. Integrate term-by-term:

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

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

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

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

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

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

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            Now substitute $u$ back in:

            $$\frac{\sin^{3}{\left(x \right)}}{3}$$

         So, the result is: $- \frac{\sin^{3}{\left(x \right)}}{3}$

      1. The integral of cosine is sine:

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

      The result is: $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$

   Method #3

   1. Rewrite the integrand:

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

   2. Integrate term-by-term:

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

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

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

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

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

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

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            Now substitute $u$ back in:

            $$\frac{\sin^{3}{\left(x \right)}}{3}$$

         So, the result is: $- \frac{\sin^{3}{\left(x \right)}}{3}$

      1. The integral of cosine is sine:

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

      The result is: $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$

3. Add the constant of integration:

   $$- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}+ \mathrm{constant}$$