Integration Strategy — Question 9

Question 9

Choose the best method for each integral. Do not evaluate. $$\int x^2e^x\,dx,\qquad \int\frac{dx}{x^2-4},\qquad \int\sqrt{9-x^2}\,dx,\qquad \int\sin^2x\,dx.$$

Question 9 – Solution

1. For $\int x^2e^x\,dx$, differentiating $x^2$ repeatedly makes it simpler, while integrating $e^x$ changes nothing. Use integration by parts twice.

2. For $\int dx/(x^2-4)$, factor $$x^2-4=(x-2)(x+2).$$ The denominator has distinct linear factors, so use partial fractions.

3. For $\int\sqrt{9-x^2}\,dx$, the form $\sqrt{a^2-x^2}$ suggests $x=a\sin\theta$. Use trigonometric substitution with $x=3\sin\theta$.

4. For $\int\sin^2x\,dx$, both sine and cosine powers are even. Use the power-reduction identity $$\sin^2x=\frac{1-\cos2x}{2}.$$

$$\boxed{\begin{array}{c|c} x^2e^x&\text{integration by parts}\\ 1/(x^2-4)&\text{partial fractions}\\ \sqrt{9-x^2}&\text{trigonometric substitution}\\ \sin^2x&\text{power reduction} \end{array}}$$