Simplify $\frac{1}{\left(\sin\left(x\right)+\cos\left(x\right)\right)^2}$ into $\frac{\csc\left(x+45\right)^2}{2}$ by applying trigonometric identities
Take the constant $\frac{1}{2}$ out of the integral
We can solve the integral $\int\csc\left(x+45\right)^2dx$ 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 $x+45$ 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 finding the derivative of the equation above
Substituting $u$ and $dx$ in the integral and simplify
The integral of $\csc(x)^2$ is $-\cot(x)$
Multiply the fraction and term in $-\left(\frac{1}{2}\right)\cot\left(u\right)$
Replace $u$ with the value that we assigned to it in the beginning: $x+45$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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