Identities Rewrite Hard Integrals

Pythagorean identities and a well-chosen triangle turn tangled roots into friendly powers of sine and cosine.

Powers and Identities

Odd powers of sine or cosine yield to substitution: peel off one factor, convert the rest with the Pythagorean identity, and the leftover derivative matches. Even powers need half-angle identities, since cosine of two theta splits a square into a constant plus a single cosine. Strategy, not memorized answers, does the work.

\sin^2 x = \frac{1-\cos 2x}{2}

Three Root Forms

Square roots of quadratics invite a substitution based on the Pythagorean identity. For a square root of (a squared minus x squared), set x = a sine theta. For (a squared plus x squared), set x = a tangent theta, and for (x squared minus a squared), use x = a secant theta. Each choice makes the root collapse.

\sqrt{a^2 - x^2} = a\cos\theta

A Worked Triangle

To integrate 1 over the square root of (4 minus x squared), set x = 2 sine theta so the root becomes 2 cosine theta and dx becomes 2 cosine theta d theta. The integral collapses to theta, and since sine theta is x over 2, the answer is arcsine of (x over 2) plus C. The triangle you draw at the end converts back to x.

\int \frac{dx}{\sqrt{4-x^2}} = \arcsin\frac{x}{2} + C

Flashcard

\sin^2\theta + \cos^2\theta = 1

Identity powering all trig substitution

This identity lets a substitution replace a quadratic expression under a root with a single squared trig factor. Rearrangements produce the tangent and secant forms too.

Check your understanding

Which substitution is appropriate for an integral containing the square root of (9 plus x squared)?

Take the unit quiz

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