MH Board · Chapter

Indefinite Integration

The concepts MH Board needs, one connected idea at a time.

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MH Board · Chapter

The concepts MH Board needs, one connected idea at a time.

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Up nextIntegration as the reverse of differentiation $\int f(x)\,dx = F(x) + c$ when $F'(x) = f(x)$ ; use the standard formulae term by term, e.g. $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c$ for $n \ne -1$ Start

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Integration as the reverse of differentiation $\int f(x)\,dx = F(x) + c$ when $F'(x) = f(x)$ ;

use the standard formulae term by term, e.g.

\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c

for

n \ne -1

Start

Rewriting before integratingExpand, divide out or use an identity first (

\sin^2 x = \frac{1 - \cos 2x}{2}

\tan^2 x = \sec^2 x - 1

) so every term matches a standard formLocked

Integration by substitutionPut $u = g(x)$ so that $g'(x)\,dx = du$ : $\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ ; in particular $\int \dfrac{f'(x)}{f(x)}\,dx = \log|f(x)| + c$ Locked
Powers and products of trigonometric functionsTurn products into sums ( $2\cos A\cos B = \cos(A + B) + \cos(A - B)$ );
The substitution t = tan(x/2)For $\int \dfrac{dx}{a + b\sin x}$ or $\dfrac{dx}{a + b\cos x}$ ,

Standard integrals of quadratic forms $\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a}$ , $\int \dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a}\log\left|\dfrac{x - a}{x + a}\right|$ , $\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac{x}{a}$ , and the rest of the tableLocked
Completing the squareWrite $ax^2 + bx + c$ as $a\left[(x + p)^2 \pm q^2\right]$ ,
Integrals of (px + q) over a quadraticWrite $px + q = A\,\dfrac{d}{dx}(ax^2 + bx + c) + B$ ;

Integration by parts $\int u\,v\,dx = u\int v\,dx - \int \left(u'\int v\,dx\right)dx$ ; choose $u$ by LIATE (log, inverse trig, algebraic, trig, exponential)Locked
Special forms from integration by parts $\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + c$ ;
Integration by partial fractionsSplit a proper rational function into $\dfrac{A}{x - a} + \dfrac{B}{x - b} + \dots$

for

\sin^m x\cos^n x

with an odd power, keep one factor and substitute for the other function

Locked

put

t = \tan\dfrac{x}{2}

\sin x = \dfrac{2t}{1 + t^2}

\cos x = \dfrac{1 - t^2}{1 + t^2}

dx = \dfrac{2\,dt}{1 + t^2}

Locked

then use the standard form with

x + p

in place of

x

Locked

the

A

part gives a log or a root, the

B

part a standard quadratic form

Locked

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

and its relatives; integrals that return to themselves, like

\int e^x\sin x\,dx

Locked

(repeated and quadratic factors too), then integrate each piece

Locked