Calculus: Differentiation, Integration and Applications
Core concepts
Ext 1 calculus extends the standard course with the product, quotient and chain rules, derivatives of $\sin$, $\cos$, $\tan$, $e^x$ and $\ln x$, and integration by substitution and by parts. You must differentiate implicitly, find equations of tangents and normals, locate stationary points and classify them, and evaluate definite integrals including areas between curves. Also know the primitives $\int \frac{f'(x)}{f(x)}dx=\ln|f(x)|+C$ and $\int f'(x)e^{f(x)}dx=e^{f(x)}+C$.
Traps
- Forgetting the inner derivative in the chain rule.
- Integration by parts sign error: $\int u,dv=uv-\int v,du$; choose $u$ to simplify (LIATE heuristic).
- Substitution requires changing the limits; otherwise substitute back before evaluating.
- Area below the axis gives a negative integral - split the interval or use absolute value for total area.
- $\ln x$ is defined only for $x>0$; use $\ln|x|$ in primitives.
Rule of thumb
For a stationary point, solve $f'(x)=0$, then use the sign of $f''$ (or a sign table) to classify: $f''>0$ minimum, $f''<0$ maximum. If $f''=0$, check signs either side.
Worked example
Find $\int x e^{x}dx$. Let $u=x$, $dv=e^x dx$, so $du=dx$, $v=e^x$. Then the integral is $xe^x-\int e^x dx=xe^x-e^x+C=e^x(x-1)+C$. Differentiating back gives $e^x(x-1)+e^x=xe^x$, confirming the answer.