Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Use partial fractions to find the indefinite integral. $$ \int \frac{3 x^{2}+3 x+1}{x\left(x^{2}+2 x+1\right)} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \( - \ln|x| + \ln|x+1|- \frac{3} {x+1} + C \).

Step 1: Separate into partial fractions

Firstly, break down the expression \(\frac{3x^{2}+3x+1}{x(x^{2} + 2x + 1)}\) into its partial fractions. The denominator can be split into \(x\) and \((x + 1)^2\), thus the expression becomes \(\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2}\). Now, set these fractions to be equal to the original expression, and solve for \(A\), \(B\), and \(C\) by comparing numerators and multiply through by the common denominator to clear it.

Step 2: Solve for A, B, and C

We equate coefficients of the powers of \(x\) on both sides to get three equations i.e. \(A=-1\), \(B=1\) and \(C=3\). Our expression thus breaks down to \(-\frac{1}{x} + \frac{1}{x+1} + \frac{3}{(x+1)^2}\).

Step 3: Integrate

Now, integrate each of these fractions. The integral of \(-\frac{1}{x}\) gives \(- \ln|x|\), the integral of \(\frac{1}{x+1}\) gives \(\ln|x+1|\), and the integral of \(\frac{3}{(x+1)^2}\) gives \(- 3 \/(x+1)\). Adding these together yields the indefinite integral of the original function.

Step 4: Write final answer

Finally, add the constant of integration, denoted by \(+C\), to the result. This gives the final solution of \( - \ln|x| + \ln|x+1|- \frac{3} {x+1} + C \).