NationMaster - Encyclopedia: Trigonometric substitution

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Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution In calculus, the integral of a function is an extension of the concept of a sum. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing the radical expressions: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In calculus, the integral of a function is an extension of the concept of a sum. ...

$1-sin^2theta;=;cos^2theta$ for $sqrt{a^2-x^2}$

$1+tan^2theta;=;sec^2theta$ for $sqrt{a^2+x^2}$

$sec^2theta-1;=;tan^2theta$ to simplify $sqrt{x^2-a^2}$

In the expression a² − x², the substitution of a sin(θ) for x makes it possible to use the identity 1 − sin²θ = cos²θ.

In the expression a² + x², the substitution of a tan(θ) for x makes it possible to use the identity tan²θ + 1 = sec²θ.

Similarly, in x² − a², the substitution of a sec(θ) for x makes it possible to use the identity sec²θ − 1 = tan²θ.

1 Examples
2 Substitutions that eliminate trigonometric functions
3 See also

Examples

Integrals containing $a 2 - x 2$

In the integral

$intfrac{dx}{sqrt{a^2-x^2}}$

one may use

$x=asin(theta) mbox{so} mbox{that} arcsin(x/a)=theta,$

$dx=acos(theta),dtheta,$

a 2 - x 2 = a 2 - a 2 sin 2 (θ) = a 2 (1 - sin 2 (θ)) = a 2 cos 2 (θ),

so that the integral becomes

$intfrac{dx}{sqrt{a^2-x^2}}=intfrac{acos(theta),dtheta}{sqrt{a^2cos^2(theta)}} =int dtheta=theta+C=arcsin(x/a)+C$

(Note that the above step requires that a > 0 and cos(θ) > 0; we can choose the a to be the positive square root of a²; and we impose the restriction on θ to be −π/2 < θ < π/2 by using the arcsin() function.)

For a definite integral, one must figure out how the bounds of integration change. For example, as x goes from 0 to a/2, then sin(θ) goes from 0 to 1/2, so θ goes from 0 to π/6. Then we have

$int_0^{a/2}frac{dx}{sqrt{a^2-x^2}} =int_0^{pi/6}dtheta=frac{pi}{6}.$

(Be careful when picking the bounds. The integration from the above section requires that −π/2 < θ < π/2, so θ going from 0 to π/6 is the only choice. If we had missed this restriction, we might have picked θ to go from π to 5π/6, which would result in the negative of the result.)

Integrals containing $a 2 + x 2$

In the integral

$intfrac{1}{a^2+x^2},dx$

one may write

$x=atan(theta) mbox{so} mbox{that} theta=arctan(x/a),$

$dx=asec^2(theta),dtheta,$

a 2 + x 2 = a 2 + a 2 tan 2 (θ) = a 2 (1 + tan 2 (θ)) = a 2 sec 2 (θ),

x / a = tan(θ),

so that the integral becomes

$intfrac{1}{a^2sec^2(theta)},asec^2(theta),dtheta =frac{1}{a}int,dtheta=frac{theta}{a}+C=frac{1}{a}arctan(x/a)+C$

(provided a > 0).

Integrals containing $x 2 - a 2$

Integrals like

$int frac{dx}{x^2 - a^2}$

should be done by partial fractions rather than trigonometric subtstitutions. In algebra, the partial fraction decomposition or (partial fraction expansion) of a rational function expresses the function as a sum of fractions, where: the denominator of each term is a power of an irreducible (not factorable) polynomial and the numerator is a polynomial of smaller degree than the denominator. ...

The integral

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

can be done by the substitution

$begin{align} x &{}= a sectheta, dx &{}= a secthetatantheta,dtheta, a^2 - x^2 &{}= a^2 tan^2theta. end{align}$

This will involve the integral of secant cubed. One of the more challenging indefinite integrals of elementary calculus is This antiderivative may be found by integration by parts, as follows: where Then Next we add to both sides of the equality just derived: Then divide both sides by 2. ...

Substitutions that eliminate trigonometric functions

Substitution can be used to remove trigonometric functions. For instance,

$int f(sin x,cos x),dx=intfrac1{pmsqrt{1-u^2}}fleft(u,pmsqrt{1-u^2}right),du, qquad qquad u=sin x$

$int f(sin x,cos x),dx=intfrac{-1}{pmsqrt{1-u^2}}fleft(pmsqrt{1-u^2},uright),du qquad qquad u=cos x$

(but be careful with the signs)

$int f(sin x,cos x),dx=intfrac2{1+u^2} fleft(frac{2u}{1+u^2},frac{1-u^2}{1+u^2}right),du qquad qquad u=tanfrac x2$

Example (see quintic of l'Hôspital):

$intfrac{cos x}{(1+cos x)^3},dx$ $=intfrac2{1+u^2}frac{frac{1-u^2}{1+u^2}}{left(1+frac{1-u^2}{1+u^2}right)^3},du$ $=frac14int(1-u^4),du$ $=frac14left(u-frac15u^5right)+C$ $=frac{(1+3cos x+cos^2x)sin x}{5(1+cos x)^3}+C$

Contents

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Integrals containing a2 − x2

Integrals containing a2 + x2

Integrals containing x2 − a2

Substitutions that eliminate trigonometric functions

See also

Examples

Integrals containing $a 2 - x 2$

Integrals containing $a 2 + x 2$

Integrals containing $x 2 - a 2$