Special / Standard Forms

MATStandard Forms & Substitution

This subtopic is a catalogue of standard integrals built around a quadratic (or its square root) in the denominator: forms reducible to $\dfrac{1}{a^2+x^2}$ , $\dfrac{1}{a^2-x^2}$ , $\dfrac{1}{\sqrt{a^2-x^2}}$ , and their relatives. The universal method is to complete the square in the quadratic, then match a memorised standard result; when a linear numerator is present you split it into (derivative of quadratic) plus a constant.

ISC awards these for clean technique, and they recur inside partial-fraction and definite-integral problems.

Reciprocal of sum / difference of squares

\int \frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C, \qquad \int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C

a>0

constant; the first gives an arctan, the second a logarithm. Complete the square first if the quadratic is not yet in this form.

Reciprocal square-root forms

\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a} + C, \qquad \int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln\left|x+\sqrt{x^2\pm a^2}\right| + C

a>0

; the first needs

|x|<a

. For

\dfrac{1}{\sqrt{5-4x-x^2}}

complete the square as

9-(x+2)^2

, an

a^2-t^2

form with

a=3

Square-root of a quadratic

\int \sqrt{a^2-x^2}\,dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C

a>0

; companion forms exist for

\sqrt{x^2\pm a^2}

, e.g.

\int\sqrt{9-x^2}\,dx

uses

a=3

Linear over quadratic — split the numerator

px+q = \frac{p}{2}\,(2x+b) + \left(q - \frac{pb}{2}\right)

For

\int\dfrac{px+q}{x^2+bx+c}\,dx

, write the numerator as

\dfrac{p}{2}\cdot(\text{derivative of denominator}) + \text{constant}

, splitting into a

\ln

part and a standard part.

Completing the square

x^2+bx+c = \left(x+\frac{b}{2}\right)^2 + \left(c - \frac{b^2}{4}\right)

Converts any quadratic into

(x+k)^2 \pm A^2

so a standard

a^2\pm x^2

form applies after the shift

t=x+\dfrac{b}{2}

Step 1 for almost every problem here: complete the square in the quadratic to reach $(x+k)^2\pm A^2$ or $A^2-(x+k)^2$ .
For a linear numerator $px+q$ , set the numerator $= \lambda(\text{denominator's derivative}) + \mu$ , solve for $\lambda,\mu$ , then integrate the two pieces separately.
The $\ln$ -piece gives $\ln|\text{quadratic}|$ (or $\dfrac{1}{2}\ln$ for square-root denominators); the constant-piece gives an arctan/arcsin/log standard form.
Distinguish $a^2+x^2$ (arctan) from $a^2-x^2$ (log) and $\sqrt{a^2-x^2}$ (arcsin) from $\sqrt{x^2\pm a^2}$ (log) — the sign and the square-root decide the answer type.
After the shift $t=x+\dfrac{b}{2}$ , the back-substitution must return to $x$ ; here $A=\sqrt{|c-b^2/4|}$ is the effective $a$ .
For $\int\dfrac{dx}{5-4x-x^2}$ , factor out the sign first: $5-4x-x^2 = 9-(x+2)^2$ , an $a^2-t^2$ form with $a=3$ .
Keep $a>0$ when quoting these results; the formulas assume a positive constant.

Where the marks go

Confusing the arctan form $\dfrac{1}{a^2+x^2}$ with the log form $\dfrac{1}{a^2-x^2}$ — a sign error flips the entire answer type.
Forgetting to factor out a leading negative when completing the square in $c-bx-x^2$ , leading to a wrong sign under the root.
Using $\sin^{-1}\dfrac{x}{a}$ when the radicand is $x^2+a^2$ (which actually gives a $\ln$ ), or vice-versa.
Splitting the linear numerator incorrectly — the coefficient of the derivative term must be $\dfrac{p}{2}$ when the derivative is $2x+b$ .

How the board asks it

Numericalcomplete the square to an arctan/log standard form
Evaluate $\displaystyle\int \dfrac{dx}{x^2 - 6x + 13}$ .
Numericalreciprocal square-root form after the shift (arcsin)
Evaluate $\displaystyle\int \dfrac{dx}{\sqrt{5 - 4x - x^2}}$ .
Numericallinear over quadratic — split numerator into derivative plus constant
Evaluate $\displaystyle\int \dfrac{2x + 3}{x^2 + 4x + 8}\,dx$ .
Numericallinear numerator over a square-root of a quadratic
Evaluate $\displaystyle\int \dfrac{x + 2}{\sqrt{x^2 + 2x + 5}}\,dx$ .
Numericaldefinite integral applying limits to a standard square-root form
Find $\displaystyle\int_{0}^{1} \dfrac{dx}{\sqrt{3 - 2x - x^2}}$ .

Practise this topic

NextIntegration by Substitution

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.