Integral of 3√(7-cos*x)*sinxdx dx

1. Let $u = 7 - \cos{\left(x \right)}$.

   Then let $du = \sin{\left(x \right)} dx$ and substitute $3 du$:

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

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

      $$\int \sqrt{u}\, du = 3 \int \sqrt{u}\, du$$

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

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

      So, the result is: $2 u^{\frac{3}{2}}$

   Now substitute $u$ back in:

   $$2 \left(7 - \cos{\left(x \right)}\right)^{\frac{3}{2}}$$

2. Add the constant of integration:

   $$2 \left(7 - \cos{\left(x \right)}\right)^{\frac{3}{2}}+ \mathrm{constant}$$

The answer is:

$$2 \left(7 - \cos{\left(x \right)}\right)^{\frac{3}{2}}+ \mathrm{constant}$$