<p>Rewrite powers as sums of integrable terms. These identities combine the AA trigonometry syllabus with SL5.10 and AHL5.15 integration.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Expression</strong></th><th style="text-align: center; --cell-justify: center"><strong>Useful identity</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\sin^2x\)</td><td style="text-align: center; --cell-justify: center">\(\dfrac{1-\cos2x}{2}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\cos^2x\)</td><td style="text-align: center; --cell-justify: center">\(\dfrac{1+\cos2x}{2}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\tan^2x\)</td><td style="text-align: center; --cell-justify: center">\(\sec^2x-1\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\cot^2x\)</td><td style="text-align: center; --cell-justify: center">\(\operatorname{cosec}^2x-1\)</td></tr></tbody></table></div></div>
<p>For an odd power of sine or cosine, reserve one factor for the differential and rewrite the remaining even power using the Pythagorean identity.</p>
<p style="text-align: center">\(\begin{aligned}\int\sin^3x\,\mathrm{d}x&=\int(1-\cos^2x)\sin x\,\mathrm{d}x\\&=-\int(1-u^2)\,\mathrm{d}u\quad(u=\cos x)\\&=-\cos x+\dfrac{\cos^3x}{3}+C\end{aligned}\)</p>
<p>Use radians. Keep the chain-rule scale factor when the angle is a multiple of the variable.</p>

Rewrite powers as sums of integrable terms. These identities combine the AA trigonometry syllabus with SL5.10 and AHL5.15 integration.

For an odd power of sine or cosine, reserve one factor for the differential and rewrite the remaining even power using the Pythagorean identity.

\(\begin{aligned}\int\sin^3x\,\mathrm{d}x&=\int(1-\cos^2x)\sin x\,\mathrm{d}x\\&=-\int(1-u^2)\,\mathrm{d}u\quad(u=\cos x)\\&=-\cos x+\dfrac{\cos^3x}{3}+C\end{aligned}\)

Use radians. Keep the chain-rule scale factor when the angle is a multiple of the variable.

Trigonometric Identities Before Integration

Trigonometric Identities Before Integration

Expression	Useful identity
\(\sin^2x\)	\(\dfrac{1-\cos2x}{2}\)
\(\cos^2x\)	\(\dfrac{1+\cos2x}{2}\)
\(\tan^2x\)	\(\sec^2x-1\)
\(\cot^2x\)	\(\operatorname{cosec}^2x-1\)