Integral of 3sin^3(x)+18sinx dx

1. Integrate term-by-term:

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

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

      1. Rewrite the integrand:

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

      2. There are multiple ways to do this integral.

         Method #1

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

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

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

            1. Integrate term-by-term:

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

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

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

                  $$\int \left(-1\right)\, du = - u$$

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

            Now substitute $u$ back in:

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

         Method #2

         1. Rewrite the integrand:

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

         2. Integrate term-by-term:

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

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

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

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

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

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

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

                     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: $- \frac{u^{3}}{3}$

                  Now substitute $u$ back in:

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

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

            1. The integral of sine is negative cosine:

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

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

         Method #3

         1. Rewrite the integrand:

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

         2. Integrate term-by-term:

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

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

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

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

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

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

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

                     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: $- \frac{u^{3}}{3}$

                  Now substitute $u$ back in:

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

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

            1. The integral of sine is negative cosine:

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

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

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

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

      $$\int 18 \sin{\left(x \right)}\, dx = 18 \int \sin{\left(x \right)}\, dx$$

      1. The integral of sine is negative cosine:

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

      So, the result is: $- 18 \cos{\left(x \right)}$

   The result is: $\cos^{3}{\left(x \right)} - 21 \cos{\left(x \right)}$

2. Now simplify:

   $$\left(\cos^{2}{\left(x \right)} - 21\right) \cos{\left(x \right)}$$

3. Add the constant of integration:

   $$\left(\cos^{2}{\left(x \right)} - 21\right) \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\left(\cos^{2}{\left(x \right)} - 21\right) \cos{\left(x \right)}+ \mathrm{constant}$$