Problem 24 - Essential Calculus

Question

Find or evaluate the integral. (Complete the square, if necessary.) $$ \int \frac{2 x-5}{x^{2}+2 x+2} d x $$

Step-by-Step Solution

Answer

The integral of \(\frac{2 x-5}{x^{2}+2 x+2} dx\) is \(2 \ln\left|\sqrt{(x+1)^2+1} \right| - 12 \arctan(x+1) + C\).

Step 1: Complete the Square

Complete the square in the denominator \(x^{2}+2 x+2\). This results in \((x+1)^2 +1\). Our integrand now becomes \(\frac{2(x-5)}{(x+1)^2+1}\).

Step 2: Substitute variable

Make a substitution \(u = x + 1\), then \(du=dx\), and \(x = u - 1\). Substitute these into the integrand: \(\frac{2(u-6)}{u^2+1} du\).

Step 3: Decompose the Fraction

We decompose the fraction into two simpler fractions by breaking it up as follows: \(\frac{2(u-6)}{u^2+1} = 2\frac{u}{u^2+1} - \frac{12}{u^2+1}\).

Step 4: Integrate Each Fraction

The integral is broken up into two integrals and each is integrated separately. This results in \(2\int \frac{u}{u^2+1} du - 12\int\frac{1}{u^2+1} du\). The first integral results in \(2 \ln\left|\sqrt{u^2+1} \right|\), and the second integral is \(-12 \arctan(u)\). So, the integral is \(2 \ln\left|\sqrt{u^2+1} \right| - 12 \arctan(u) + C\), where C is the constant of integration.

Step 5: Substitute u back in

Substitute \(u = x+1\) back into our equation. Our final answer is \(2 \ln\left|\sqrt{(x+1)^2+1} \right| - 12 \arctan(x+1) + C\).