Final Answer
Step-by-step explanation
Problem to solve:
Choose the solving method
We can solve the integral $\int\frac{1}{2\sin\left(x\right)\cos\left(x\right)}dx$ 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 $\int\frac{\frac{1}{2}\left(1+t^{2}\right)}{t\left(1-t^{2}\right)}dt$ 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
Rewriting $t$ in terms of $u$
Substituting $u$, $dt$ and $t$ in the integral and simplify
Rewrite the fraction $\frac{\frac{1}{4}u}{\left(u-1\right)\left(2-u\right)}$ in $2$ simpler fractions using partial fraction decomposition
Find the values for the unknown coefficients: $A, B$. The first step is to multiply both sides of the equation from the previous step by $\left(u-1\right)\left(2-u\right)$
Multiplying polynomials
Simplifying
Expand the polynomial
Assigning values to $u$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{\frac{1}{4}u}{\left(u-1\right)\left(2-u\right)}$ in decomposed fraction equals
The integral of the sum of two or more functions is equal to the sum of their integrals
We can solve the integral $\int\frac{\frac{1}{4}}{u-1}du$ 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 $v$), which when substituted makes the integral easier. We see that $u-1$ it's a good candidate for substitution. Let's define a variable $v$ and assign it to the choosen part
Now, in order to rewrite $du$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by deriving the equation above
Substituting $v$ and $du$ in the integral and simplify
The integral $\int\frac{\frac{1}{4}}{v}dv$ results in: $\frac{1}{2}\ln\left(t\right)$
The integral $\int\frac{\frac{1}{2}}{2-u}du$ results in: $-\frac{1}{2}\ln\left|2-\left(1+t^{2}\right)\right|$
Gather the results of all integrals
Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$