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Mathematics and Statistics · Ch 5 — Integration

Integrals of Standard Forms

4

Integrals of Standard Forms

A group of frequently-occurring integrals involving x2±a2x^{2}\pm a^{2} have fixed results worth memorising. Each is derived once (by substitution or partial fractions) and then quoted directly.

The standard-form results

#IntegralResult
1∫dxx2−a2\displaystyle\int \frac{dx}{x^{2}-a^{2}}12alog⁡∣x−ax+a∣+c\dfrac{1}{2a}\log\left\lvert\dfrac{x-a}{x+a}\right\rvert+c
2∫dxa2−x2\displaystyle\int \frac{dx}{a^{2}-x^{2}}12alog⁡∣a+xa−x∣+c\dfrac{1}{2a}\log\left\lvert\dfrac{a+x}{a-x}\right\rvert+c
3∫dxx2+a2\displaystyle\int \frac{dx}{x^{2}+a^{2}}1atan⁡−1 ⁣(xa)+c\dfrac{1}{a}\tan^{-1}\!\left(\dfrac{x}{a}\right)+c
4∫dxx2+a2\displaystyle\int \frac{dx}{\sqrt{x^{2}+a^{2}}}log⁡∣x+x2+a2∣+c\log\left\lvert x+\sqrt{x^{2}+a^{2}}\right\rvert+c
5∫dxx2−a2\displaystyle\int \frac{dx}{\sqrt{x^{2}-a^{2}}}log⁡∣x+x2−a2∣+c\log\left\lvert x+\sqrt{x^{2}-a^{2}}\right\rvert+c
6∫dxa2−x2\displaystyle\int \frac{dx}{\sqrt{a^{2}-x^{2}}}sin⁡−1 ⁣(xa)+c\sin^{-1}\!\left(\dfrac{x}{a}\right)+c

Reading the forms correctly

The deciding feature is always which term is subtracted from which, and whether a square root is present:

  • x2−a2x^{2}-a^{2} in the denominator (no root) uses form 1; a2−x2a^{2}-x^{2} (no root) uses form 2 — the log's fraction flips accordingly.
  • x2+a2x^{2}+a^{2} (no root) gives an inverse-tangent (form 3).
  • A square root in the denominator sends you to forms 4–6; a plus sign under the root and an x2−a2x^{2}-a^{2} give the log-forms 4/5, while a2−x2a^{2}-x^{2} under the root gives the inverse-sine (form 6).

Verifying form 1 by differentiation

Using log⁡∣u/v∣=log⁡∣u∣−log⁡∣v∣\log\lvert u/v\rvert=\log\lvert u\rvert-\log\lvert v\rvert and the chain rule, …

Definition 1Log standard forms

∫dxx2−a2=12alog⁡∣x−ax+a∣+c\int\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+c and ∫dxa2−x2=12alog⁡∣a+xa−x∣+c\int\frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+c — used to identify aa fr …

Definition 2Root and inverse-function forms

∫dxx2+a2=1atan⁡−1xa+c\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}+c; ∫dxx2±a2=log⁡∣x+x2±a2∣+c\int\frac{dx}{\sqrt{x^{2}\pm a^{2}}}=\log|x+\sqrt{x^{2}\pm a^{2}}|+c; $\int\frac{dx}{\sqrt{a^{2}-x^ …

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