Integration Strategy — Question 10

Question 10

Find the integrand whose antiderivative is $$F(x)=\ln(x^2+1)+\arctan x.$$ Then verify your result.

Question 10 – Solution

Step 1: Differentiate the logarithm using the chain rule. $$\frac{d}{dx}\ln(x^2+1)=\frac{2x}{x^2+1}.$$ Step 2: Differentiate the inverse tangent. $$\frac{d}{dx}\arctan x=\frac1{x^2+1}.$$ Step 3: Add and simplify. $$\begin{align*} F'(x)&=\frac{2x}{x^2+1}+\frac1{x^2+1}\\ &=\frac{2x+1}{x^2+1}. \end{align*}$$ Thus the required integrand is $$\boxed{f(x)=\frac{2x+1}{x^2+1}}$$ Step 4: Verify by integration. $$\begin{align*} \int\frac{2x+1}{x^2+1}\,dx &=\int\frac{2x}{x^2+1}\,dx+\int\frac{dx}{x^2+1}\\ &=\ln(x^2+1)+\arctan x+C=F(x)+C. \end{align*}$$