Final answer to the problem
Step-by-step Solution
Specify the solving method
Simplifying
We can solve the integral $\int\frac{\sin\left(x\right)}{1+\cos\left(x\right)}dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution
Hence
Substituting in the original integral we get
Simplifying
We can solve the integral $\int_{\frac{3\pi}{2}}^{2\pi }\frac{2t}{1+t^{2}}dt$ 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 $1+t^{2}$ 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 $dt$ 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 $dt$ in the previous equation
Substituting $u$ and $dt$ 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: $1+t^{2}$
Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$
Evaluate the definite integral
Simplify the expression inside the integral