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Substitutions for

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

and

\sqrt{x^2-a^2}

MTCAL 1Integration techniques

For $a>0$ , choose the indicated trigonometric or hyperbolic branch so that the substitution is one-to-one on the interval being integrated. Keep the absolute value when simplifying a square root until the chosen range fixes its sign; for example, $\sqrt{a^2-a^2\sin^2\theta}=a|\cos\theta|$ .

Integrand factor	Substitutions	Identities
$\sqrt{a^2-x^2}$	$x=a\sin\theta$ , $dx=a\cos\theta\,d\theta$ or $x=a\tanh\theta$ , $dx=a\operatorname{sech}^2\theta\,d\theta$	$\cos^2\theta+\sin^2\theta=1$ , $\operatorname{sech}^2\theta+\tanh^2\theta=1$
$\sqrt{a^2+x^2}$	$x=a\tan\theta$ , $dx=a\sec^2\theta\,d\theta$ or $x=a\sinh\theta$ , $dx=a\cosh\theta\,d\theta$	$\sec^2\theta=\tan^2\theta+1$ , $\cosh^2\theta=1+\sinh^2\theta$
$\sqrt{x^2-a^2}$	$x=a\sec\theta$ , $dx=a\sec\theta\tan\theta\,d\theta$ or, for $x\ge a$ , $x=a\cosh\theta$ , $dx=a\sinh\theta\,d\theta$	$\sec^2\theta=1+\tan^2\theta$ , $\cosh^2\theta=1+\sinh^2\theta$

5Accumulation and changes of variables / single integrals

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The substitution

y=xv

MTCAL 2Ordinary differential equations

For an equation of the form

\frac{dy}{dx}=f\left(\frac{y}{x}\right),

set $y=xv(x)$ . Then

y'=v+xv',

xv'=f(v)-v, \qquad \frac{dv}{dx}=\frac{f(v)-v}{x}.

The resulting equation is separable on an interval where $x\neq0$ and the required denominators are nonzero.

5Accumulation and changes of variables / ODEs: a substitution makes it separable

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Diagonalisation

DFALG 4Eigenvalues

An $n\times n$ matrix $A$ over a field $F$ is diagonalisable over $F$ if there are an invertible matrix $P\in M_n(F)$ and a diagonal matrix $D\in M_n(F)$ such that

P^{-1}AP=D, \qquad\text{equivalently}\qquad A=PDP^{-1}.

The columns of $P$ are the eigenvectors in the chosen eigenbasis and the corresponding eigenvalues appear on the diagonal of $D$ .

5Accumulation and changes of variables / linear maps: a basis that decouples

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Change of variables

THCAL 5Double integrals

Let $\Omega$ be a region in $(x,y)$ -coordinates and let $\Phi$ be a region in $(u,v)$ -coordinates. Assume that $f$ is integrable on $\Omega$ , that $g$ is $C^1$ on $\Phi$ , that $\Phi$ can be split into finitely many pieces on which $g$ is one-to-one with non-overlapping interiors and that the Jacobian is nonzero away from boundary or other exceptional sets of area zero. The images of those pieces must cover $\Omega$ once, apart from boundaries of area zero; otherwise the transformed integral can omit or count part of the region more than once. If a map covers the target with multiplicity, split the parameter region into one-to-one pieces (or account for that multiplicity) before applying the formula. Then

\iint_\Omega f(x,y)\,dx\,dy = \iint_\Phi f\bigl(x(u,v),y(u,v)\bigr) \left|\det J_g(u,v)\right|\,du\,dv.

5Accumulation and changes of variables / double integrals: the Jacobian scales area

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Polar area element

THCAL 5Double integrals

For $r\ge0$ , the polar-coordinate change of variables is

x=r\cos\theta,\qquad y=r\sin\theta,

and its Jacobian satisfies

\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right| =r.

Consequently,

dA=r\,dr\,d\theta,

and

\iint_\Omega f(x,y)\,dA =\iint_\Phi f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta.

5Accumulation and changes of variables / polar coordinates

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