Integrals Involving Quadratics — Question 2

Question 2

Split the numerator, then evaluate: $$\int\frac{3x+1}{x^2+2x+5}\,dx.$$

Question 2 – Solution

Step 1: Split the numerator using the derivative of the denominator. Since $$\frac{d}{dx}(x^2+2x+5)=2x+2,$$ write $$3x+1=\frac32(2x+2)-2.$$ Step 2: Split the integral. $$\begin{align*} I={}&\frac32\int\frac{2x+2}{x^2+2x+5}\,dx -2\int\frac{dx}{x^2+2x+5}. \end{align*}$$ Step 3: Integrate the derivative term. $$\frac32\int\frac{2x+2}{x^2+2x+5}\,dx =\frac32\ln(x^2+2x+5).$$ Step 4: Complete the square in the remaining term. $$x^2+2x+5=(x+1)^2+4.$$ Thus $$\begin{align*} -2\int\frac{dx}{(x+1)^2+2^2} &=-2\left[\frac12\arctan\left(\frac{x+1}{2}\right)\right]\\ &=-\arctan\left(\frac{x+1}{2}\right). \end{align*}$$ Step 5: Combine the results. $$\boxed{\frac32\ln(x^2+2x+5)-\arctan\frac{x+1}{2}+C}$$