Integrals Involving Quadratics — Question 9

Question 9

Split the numerator, then evaluate: $$\int\frac{2x+5}{x^2+4x+8}\,dx.$$

Question 9 – Solution

Step 1: Split the numerator. $$2x+5=(2x+4)+1.$$ Step 2: Split the integral. $$\begin{align*} I={}&\int\frac{2x+4}{x^2+4x+8}\,dx +\int\frac{dx}{x^2+4x+8}. \end{align*}$$ Step 3: Integrate the derivative term. $$\int\frac{2x+4}{x^2+4x+8}\,dx=\ln(x^2+4x+8).$$ Step 4: Complete the square for the remaining term. $$x^2+4x+8=(x+2)^2+4.$$ Therefore, $$\int\frac{dx}{(x+2)^2+2^2} =\frac12\arctan\left(\frac{x+2}{2}\right).$$ Step 5: Combine the results. $$\boxed{\ln(x^2+4x+8)+\frac12\arctan\frac{x+2}{2}+C}$$