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Trigonometric Substitution

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Integrals of the form

intf(costheta,sintheta)dtheta
(1)

can be solved by making the substitution z=e^(itheta) so that dz=ie^(itheta)dtheta and expressing

costheta=(e^(itheta)+e^(-itheta))/2
(2)
=(z+z^(-1))/2
(3)
sintheta=(e^(itheta)-e^(-itheta))/(2i)
(4)
=(z-z^(-1))/(2i).
(5)

The integral can then be solved by contour integration.

Alternatively, making the Weierstrass substitution t=tan(theta/2) transforms (◇) into

intf((1-t^2)/(1+t^2),(2t)/(1+t^2))(2dt)/(1+t^2).
(6)

The following table gives trigonometric substitutions which can be used to transform integrals involving square roots.

formsubstitution
sqrt(a^2-x^2)x=asintheta
sqrt(a^2+x^2)x=atantheta
sqrt(x^2-a^2)x=asectheta

See also

Contour Integration, Hyperbolic Substitution, Integral, Integration, Weierstrass Substitution

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Cite this as:

Weisstein, Eric W. "Trigonometric Substitution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometricSubstitution.html

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