Further calculus (Further Pure 2)

Explanation

• A reduction formula expresses an integral $I_n$ that depends on a positive integer $n$ in terms of $I_{n-1}$ or $I_{n-2}$, so that repeated use drives the index down to a base case you can integrate directly. The derivation is almost always integration by parts, and the skill is choosing the split.
• For $\int x^{n}f(x)\,\mathrm{d}x$ take $u=x^{n}$ so that differentiating lowers the power.
• For $\int\sin^{n}x\,\mathrm{d}x$ write the integrand as $\sin^{n-1}x\times\sin x$, integrate the second factor and then replace $\cos^{2}x$ by $1-\sin^{2}x$; the resulting equation contains $I_n$ on both sides and is rearranged.
• For $\int\tan^{n}x\,\mathrm{d}x$ split off $\tan^{2}x=\sec^{2}x-1$ instead, which needs no parts at all.
• Always state the base case, $I_0$ or $I_1$, and apply the formula in descending steps, showing each one.

Explanation

• For a curve $y=f(x)$ the arc length between $x=a$ and $x=b$ is $s=\displaystyle\int_{a}^{b}\sqrt{1+\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}}\,\mathrm{d}x$.
• In parametric form it is $s=\displaystyle\int_{t_1}^{t_2}\sqrt{\left(\dfrac{\mathrm{d}x}{\mathrm{d}t}\right)^{2}+\left(\dfrac{\mathrm{d}y}{\mathrm{d}t}\right)^{2}}\,\mathrm{d}t$, and in polar form $s=\displaystyle\int_{\alpha}^{\beta}\sqrt{r^{2}+\left(\dfrac{\mathrm{d}r}{\mathrm{d}\theta}\right)^{2}}\,\mathrm{d}\theta$.
• Rotating the arc about the $x$-axis produces a surface of area $S=2\pi\displaystyle\int y\,\mathrm{d}s$, so $S=2\pi\displaystyle\int_{a}^{b}y\sqrt{1+\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}}\,\mathrm{d}x$, with the parametric analogue obtained by replacing $\mathrm{d}s$.
• Almost every examinable curve is chosen so that $1+\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}$ collapses to a perfect square or to a standard hyperbolic identity, so simplify under the root before integrating.