Trigonometric Integrals and Trigonometric Substitution
Handle powers of sine and cosine with identities, and clear expressions such as √(a² - x²) by substituting a trigonometric function for x.
What a learner can do afterwards
- Reduce an odd power of sine by splitting off one factor and substituting
- Choose the substitution that matches a sum or difference of squares
- Convert the answer back to the original variable using a right triangle
1 · Read
Powers of sine and cosine fall to one identity: sin^2 x + cos^2 x = 1. When a power is odd, split off a single factor to serve as du, convert the rest with the identity, and substitute. When the powers are even, half-angle formulas lower the degree instead.
Take the integral of sin^3 x dx. Split it as sin^2 x sin x and convert: (1 - cos^2 x) sin x. Set u = cos x, so du = -sin x dx. The integral becomes an easy polynomial in u. For cos^5 x you mirror it: split off one cos x and set u = sin x.
Roots of squares invite a trig substitution, and the root shape picks it. Use x = a sin theta for sqrt(a^2 - x^2), x = a tan theta for sqrt(a^2 + x^2), and x = a sec theta for sqrt(x^2 - a^2). Convert dx too, such as dx = a cos theta d theta, integrate in theta, then draw a right triangle to return to x.
Matching the root shape is the whole decision: sqrt(16 - x^2) wants x = 4 sin theta. If theta stands alone at the end, write it as sin^-1(x/a). Never skip the trip back to x.
Split odd powers, halve even ones, and match each root to its trig substitution.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.