Final Answer
Step-by-step Solution
Specify the solving method
Rewrite the trigonometric expression $\frac{\sin\left(x\right)^3}{\cos\left(x\right)}$ inside the integral
We can solve the integral $\int\frac{\left(1-\cos\left(x\right)^2\right)\sin\left(x\right)}{\cos\left(x\right)}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\cos\left(x\right)$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Expand the fraction $\frac{1-u^2}{u}$ into $2$ simpler fractions with common denominator $u$
Simplify the resulting fractions
Expand the integral $\int\left(\frac{1}{u}-u\right)du$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $-\int\frac{1}{u}du$ results in: $-\ln\left(\cos\left(x\right)\right)$
The integral $-\int-udu$ results in: $\frac{1}{2}\cos\left(x\right)^2$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$