Rewrite the expression $\frac{5-x}{2x^2+x-1}$ inside the integral in factored form
Take the constant $\frac{1}{2}$ out of the integral
We can solve the integral $\int\frac{5-x}{\left(x+\frac{1}{4}\right)^2-\frac{9}{16}}dx$ 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 $x+\frac{1}{4}$ 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
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
Expand the fraction $\frac{\frac{21}{4}-u}{u^2-\frac{9}{16}}$ into $2$ simpler fractions with common denominator $u^2-\frac{9}{16}$
Simplify the expression
The integral $\frac{1}{2}\int\frac{\frac{21}{4}}{u^2-\frac{9}{16}}du$ results in: $-\frac{7}{4}\ln\left(\frac{4\left(x+\frac{1}{4}\right)+3}{4x-2}\right)$
The integral $-\frac{1}{2}\int\frac{u}{u^2-\frac{9}{16}}du$ results in: $-\frac{1}{4}\ln\left(\left(x+\frac{1}{4}\right)^2-\frac{9}{16}\right)$
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 $C$
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