Final Answer
Step-by-step Solution
Specify the solving method
Simplifying
We can solve the integral $\int_{\frac{3\pi}{2}}^{2\pi }\frac{\sin\left(x\right)}{1+\cos\left(x\right)}dx$ 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 $1+\cos\left(x\right)$ 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
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Replace $u$ with the value that we assigned to it in the beginning: $1+\cos\left(x\right)$
Evaluate the definite integral
Simplify the expression inside the integral