Problem 25 - 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 is \(\ln |(x+1)^{2}+1| - 5 \arctan(x+1) + C\).

Step 1: Rewrite the integral

Rewrite the integral as \(\int \frac{2x}{x^{2}+2x+2}dx - \int \frac{5}{x^{2}+2x+2}dx\) to divide the two terms to facilitate the next steps.

Step 2: Complete the square in the denominator

Complete the square in the denominator of each fractional part to make it easier to integrate. The denominator \(x^{2} + 2x + 2\) can be written in the form \((x+1)^{2}+1\). Now our integral looks like this: \(\int \frac{2x}{(x+1)^{2}+1}dx - \int \frac{5}{(x+1)^{2}+1}dx\).

Step 3: Find the integral of each term

Next, we integrate each term separately. For the first term: The substitution \(u = x+1\) simplifies the denominator to \(u^{2}+1\), this is recognized as the integral of tan, and results in \(\int \frac{2u}{u^{2}+1}du = \ln |u^{2}+1|\). For the second term: This turns out to be the integral form of arctan(x), resulting in \(\int \frac{5}{u^{2}+1}du = 5 \arctan(u)\). Be sure to change back the variable of integration from u to x in your final answer.

Step 4: Combine the results

Final step is to combine the results from step 3. We get \(\ln |(x+1)^{2}+1| - 5\arctan(x+1) + C\), where \(C\) represents the constant of integration.