Indefinite Integration Notes

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Indefinite IntegrationIntegration as anti-derivative

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After the notes

Formula sheet

30 results from this chapter, grouped the way the notes teach them.

Standard integrals

$\int x^n\,dx$: $\dfrac{x^{n+1}}{n+1},\ \ n\ne-1$
$\int \dfrac{dx}{x}$: $\ln|x|$
$\int e^x\,dx,\ \ \int a^x\,dx$: $e^x,\ \ \dfrac{a^x}{\ln a}$
$\int \tan x\,dx,\ \ \int \cot x\,dx$: $\ln|\sec x|,\ \ \ln|\sin x|$
$\int \sec x\,dx$: $\ln|\sec x+\tan x|$
$\int \csc x\,dx$: $\ln|\csc x-\cot x|$
$\int f(ax+b)\,dx$ , if $\int f(x)\,dx=F(x)$: $\dfrac1a F(ax+b)$

Quadratic forms

$\int \dfrac{dx}{x^2+a^2}$: $\dfrac1a\tan^{-1}\dfrac xa$
$\int \dfrac{dx}{x^2-a^2}$: $\dfrac1{2a}\ln\left|\dfrac{x-a}{x+a}\right|$
$\int \dfrac{dx}{a^2-x^2}$: $\dfrac1{2a}\ln\left|\dfrac{a+x}{a-x}\right|$
$\int \dfrac{dx}{\sqrt{a^2-x^2}}$: $\sin^{-1}\dfrac xa$
$\int \dfrac{dx}{x\sqrt{x^2-a^2}}$: $\dfrac1a\sec^{-1}\dfrac xa$
$\int \dfrac{dx}{\sqrt{x^2\pm a^2}}$: $\ln\left|x+\sqrt{x^2\pm a^2}\right|$
$\int \sqrt{a^2-x^2}\,dx$: $\dfrac x2\sqrt{a^2-x^2}+\dfrac{a^2}2\sin^{-1}\dfrac xa$
$\int \sqrt{x^2\pm a^2}\,dx$: $\dfrac x2\sqrt{x^2\pm a^2}\pm\dfrac{a^2}2\ln\left|x+\sqrt{x^2\pm a^2}\right|$
$\int\dfrac{dx}{\text{quadratic}}$ : discriminant $D<0$ / $D>0$: $\tan^{-1}$ form / $\ln$ form (complete the square)

Substitution and by parts

$\sqrt{a^2-x^2}$ , $\sqrt{a^2+x^2}$ , $\sqrt{x^2-a^2}$: $x=a\sin\theta$ , $x=a\tan\theta$ , $x=a\sec\theta$
$\sqrt{\dfrac{a^2-x^2}{a^2+x^2}}$: $x^2=a^2\cos2\theta$
$\int \dfrac{f'(x)}{f(x)}\,dx$: $\ln|f(x)|$
$\int [f(x)]^n f'(x)\,dx$: $\dfrac{[f(x)]^{n+1}}{n+1},\ \ n\ne-1$
Integration by parts: $\int uv\,dx=u\int v\,dx-\int\left(u'\int v\,dx\right)dx$
Choosing the first function: ILATE: Inverse trig, Log, Algebraic, Trig, Exponential
$\int e^x\big(f(x)+f'(x)\big)\,dx$: $e^x f(x)$
$\int \big(f(x)+xf'(x)\big)\,dx$: $xf(x)$
$\int e^x\,p(x)\,dx$ , $p$ a polynomial of degree $n$: $e^x q(x)$ , $q$ of degree $n$ : differentiate and equate coefficients

Trigonometric types

$\int\dfrac{dx}{a+b\sin^2x}$ , $\dfrac{dx}{a\sin^2x+b\cos^2x+c}$: Multiply num. and den. by $\sec^2x$ , put $\tan x=t$
$\int\dfrac{dx}{a+b\sin x}$ , $\dfrac{dx}{a\sin x+b\cos x+c}$: Put $\tan\dfrac x2=t$
$\int\dfrac{a\sin x+b\cos x+c}{l\sin x+m\cos x+n}\,dx$: Num. $=A\cdot$ Den. $+B\cdot\dfrac{d}{dx}$ (Den.) $+C$
$\int\dfrac{x^2\pm1}{x^4+kx^2+1}\,dx$: Divide by $x^2$ ; put $x\mp\dfrac1x=t$
$\sin x\pm\cos x$ in the numerator, $\sin2x$ in the denominator: Write $\sin2x=(\sin x+\cos x)^2-1$ or $1-(\sin x-\cos x)^2$

Last look

Quick revision

The checks to run before you sit a question on functions.

1Simplify first: try to bring the integrand into, or very close to, one of the standard forms.
2Never forget $+C$ , and keep $|\ |$ inside every $\ln$ .
3If the derivative of one part sits next to it, substitute that part — $\int f(g(x))\,g'(x)\,dx$ .
4For $\tan10x\tan7x\tan3x$ -type products, use $\tan A\tan B\tan(A+B)=\tan(A+B)-\tan A-\tan B$ .
5By parts: pick the first function with ILATE; when only one function is visible, take $1$ as the second.
6Seeing $e^x$ times a sum? Check whether it splits as $f+f'$ before doing by parts.
7Partial fractions need a proper fraction — divide first if the degree of the numerator is not lower.
8Linear over quadratic: write the numerator as $A\cdot\frac{d}{dx}(\text{den.})+B$ .
9If an integral is said to be rational, the coefficients of the $\frac1{x-a}$ terms in its partial fractions must be $0$ .
10For $\int\sqrt{\tan x}$ or $\sqrt{\cot x}$ , put $\tan x=y^2$ and split into $\frac{y^2+1}{y^4+1}$ and $\frac{y^2-1}{y^4+1}$ .