Integration Strategy — Question 5

Question 5

Recognize a product derivative and evaluate: $$\int e^x(x+1)\,dx.$$

Question 5 – Solution

Step 1: Look for a product whose derivative matches the integrand. Try $xe^x$.

Step 2: Differentiate with the product rule. $$\begin{align*} \frac{d}{dx}(xe^x) &=\frac{d}{dx}(x)e^x+x\frac{d}{dx}(e^x)\\ &=e^x+xe^x\\ &=e^x(x+1). \end{align*}$$ Step 3: Reverse the derivative. $$\int e^x(x+1)\,dx=xe^x+C.$$ $$\boxed{xe^x+C}$$