Integration Strategy — Question 1

Question 1

Choose the shortest method and evaluate: $$\int xe^{x^2}\,dx.$$

Question 1 – Solution

Step 1: Identify the inner function. The exponent is $x^2$, and its derivative $2x$ is present except for a constant factor.

Step 2: Substitute $u=x^2$. Then $$du=2x\,dx,\qquad x\,dx=\frac12du.$$ Step 3: Rewrite and integrate. $$\begin{align*} \int xe^{x^2}\,dx &=\frac12\int e^u\,du\\ &=\frac12e^u+C. \end{align*}$$ Step 4: Return to $x$. $$\boxed{\frac12e^{x^2}+C}$$