


IntegrationIntegration by Trigonometric Substitution IWe assume that you are familiar with the material in integration by substitution 1 and integration by substitution 2 and inverse trigonometric functions. This page will use three notations interchangeably, that is, arcsin z, asin z and sin^{1}z all mean the inverse of sin z When an integrand contains x^{2} + k^{2} but there is no way to obtain an x for replacing x dx by du, we may be able to use the trig identity, 1 + tan^{2}x = sec^{2}x or more generally k^{2} + (ktan x)^{2} = (ksec x)^{2} When an integrand contains x^{2} − k^{2}, we may be able to use the trig identity, sin^{2}x + cos^{2}x = 1 that is k^{2} − (ksin x)^{2} = (kcos x)^{2}
ExamplesConsider the integral . Note that substituting g(x) = x^{2} + 1 by u will not work, as g '(x) = 2x is not a factor of the integrand. Let us make the substitution x = tan θ then and dx = sec^{2} θ dθ. The integral becomes
Consider this integral Substitute x = sin θ then dx = cos θ dθ. Solution of the integral becomes
Now a little more complex example: In order to use the first identity, we need 4x^{2} = 9tan^{2}p. Solving for x gives x = tan p. Hence dx = sec^{2}p dp and, rearranging again, p = arctan(). Substituting, simplifying, integrating and resubstituting gives:
ExerciseTry some of these. Now some closely related examples to point out the importance of signs and roots in substitutions. ExamplesWe want x^{2} = 3sin^{2}u so we can use an identity. Let x = sin u and then dx = cos u du. Substituting, simplifying, integrating and resubstituting gives:
We want x^{2} = 3tan^{2}u so we can use an identity. Let x = tan u and then dx = sec^{2}u du. Substituting, simplifying, integrating and resubstituting gives:
We need x^{2} = 3tan^{2}u so we can substitute. Let x = tan u and then dx = sec^{2}u du. Substituting, simplifying, integrating and resubstituting gives: This integral is apparently simpler but is beyond the integration tools covered so far.
We can try x^{2} = 3sin^{2}u. Let x = sin u and then dx = cos^{2}u du. Substituting, simplifying, integrating and resubstituting gives: This problem is better approached with the clever use of some High School algebra. By splitting the reciprocal of a difference of two squares into a simpler pair of fractions we obtain an integrable expression. This is an example of the method of partial fractions. (The keen might want to show this by simplifying the right side. Show me now.)
Then the integration is simpler:
ExerciseYou should do plenty of exercises until you are sure you recognise each type of problem and its solution. Look now at the more general situation where you have a fraction with a numeric term divided by a quadratic expression. If you are able to express the quadratic expression in the form (linear expression) squared plus some number, the tan substitution is possible. ExampleFirst see whether the quadratic fits the pattern by completing the square. x^{2} + 12x + 45 = (x + 6)^{2} + 9. It does. Set x = 3tan w − 6. then dx = 3sec^{2} w and w = arctan().
Alternatively, if the expression is a fraction with a numeric term divided by the square root of a quadratic expression and the quadratic can be expressed as a positive number minus some linear expression squared, a sin substitution is possible. ExampleFirst see whether the quadratic fits the pattern by completing the square. 9 + 8x − x^{2} = 25 − (x − 4)^{2}. It does. Substitute x = 5sin w + 4 , then dx = 5cos ^{} w and w = arcsin().
ExercisesDecide whether trigonometric substitution will be helpful for these expressions and integrate them if possible. << Integration by Algebraic Substitution 2  Integration Index  Integration by Trigonometric Substitution 2 >> 