Integration (Pure 3)

The integral that gives you angles

• You can integrate $\dfrac{1}{x^2+1}$ to get $\tan^{-1}x$ — a function that returns angles.
• This connection between algebra and geometry is one of the most beautiful results in calculus.

New standard integrals 标准积分

• A new standard integral: $\displaystyle\int \frac{1}{x^2 + a^2}\,dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$.
• The pattern $\dfrac{k\,f'(x)}{f(x)}$ → $k\ln|f(x)| + C$, e.g. $\displaystyle\int \frac{2x}{x^2+1}\,dx = \ln(x^2+1) + C$.

The locus |z - a| = r is a circle of radius r centred at a

Partial fractions 部分分式 for integration

• Partial fractions: split a rational function, then integrate each piece as a logarithm.
• Example: $\displaystyle\int \dfrac{1}{x^2-1}\,dx = \int\left(\dfrac{1}{2(x-1)} - \dfrac{1}{2(x+1)}\right)dx = \dfrac{1}{2}\ln|x-1| - \dfrac{1}{2}\ln|x+1| + C$.

Integration by parts 分部积分

• Integration by parts (for a product): $\displaystyle\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$.
• Choose $u$ to be the part that simplifies when differentiated.

Integration by substitution 换元积分

• Integration by substitution: a change of variable 变量替换 turns a hard integral into an easy one.
• Let $u = g(x)$, then $du = g'(x)\,dx$, and rewrite the integral in terms of $u$.

allow_no_figure: Integration techniques (by parts, substitution, partial fractions) — the visual content is the algebraic manipulation, not a diagram.

• Integration is reverse differentiation.

You've got it

Write from a equals 5

• Integrate 1 over x squared plus 25.