Integration by Recognition (Ex 11H)
Year 12 Mathematical Methods Unit 3 · Cambridge §11H · Mr Wong
Learning intentions.
- Differentiate a given function and rearrange to read off a related antiderivative
- Handle the missing scalar that almost always appears (e.g. $\tfrac{1}{k}$ out the front)
- Apply recognition to product-rule integrals like $\int x\cos(kx)\,dx$ and $\int xe^{kx}\,dx$
Key rule
If $\dfrac{d}{dx}\bigl[F(x)\bigr] = g(x)$ then $\displaystyle\int g(x)\,dx = F(x) + c$.
Differentiate the given function, then compare to the target integrand. Pull out the scalar needed to match. Always sanity-check by differentiating your final answer.
Part A — Chain-rule recognition
1. Let $f(x) = \log_e(x^3 + 2)$.
a Find $f'(x)$.
b Hence find $\displaystyle\int \dfrac{x^2}{x^3+2}\,dx$.
2. Let $f(x) = \log_e(3x^2 + 7)$.
a Find $f'(x)$.
b Hence evaluate $\displaystyle\int_0^2 \dfrac{x}{3x^2+7}\,dx$ exactly.
3. Let $f(x) = e^{x^2+1}$.
a Find $f'(x)$.
b Hence find $\displaystyle\int xe^{x^2+1}\,dx$.
4. Let $f(x) = \sin(x^2)$. Find $f'(x)$, then find $\displaystyle\int x\cos(x^2)\,dx$.
Integration by Recognition — continued
Part B (product-rule recognition) · VCAA-style application
Part B — Product-rule recognition
5. Let $f(x) = x\sin(3x)$.
a Find $f'(x)$.
b Hence find $\displaystyle\int_0^{\pi/6} x\cos(3x)\,dx$ in exact form.
6. Let $f(x) = xe^x$. Find $f'(x)$, hence find $\displaystyle\int xe^x\,dx$.
7. Let $f(x) = x^2 e^x$. Find $f'(x)$, hence find $\displaystyle\int (x^2 + 2x)e^x\,dx$.
8. Let $f(x) = (x^2 + 1)^5$. Find $f'(x)$, hence find $\displaystyle\int x(x^2 + 1)^4\,dx$.
Part C — VCAA-style
9. VCAA 2013 Exam 1 (Tech-Free, adapted).
Let $g(x) = \dfrac{x}{\sqrt{x^2+1}}$.
a Show that $\dfrac{d}{dx}\bigl[\sqrt{x^2+1}\bigr] = g(x)$.
b Hence evaluate $\displaystyle\int_0^{\sqrt{3}} \dfrac{x}{\sqrt{x^2+1}}\,dx$.