<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>\[\frac{\mathrm{d}}{\mathrm{d}x}\left( \ln x \right)=\frac{1}{x}\]</p></div><div class="question-column-divider"></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>\[\frac{\mathrm{d}}{\mathrm{d}x}\left( \sqrt{x} \right)=\frac{1}{2\sqrt{x}}\]</p></div></div></div>

Use an identity to rewrite each squared trigonometric function as a sum of standard integrable terms.

Identity check

Differentiate the result to check the identity and signs.

Integrating Powers of Sine and Cosine

Techniques of Integration

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Integral	Identity and rewrite	Antiderivative
\(\displaystyle\int\sin^2x\,\mathrm{d}x\)	\(\sin^2x=\dfrac{1-\cos2x}{2}\)	\(\dfrac{x}{2}-\dfrac{\sin2x}{4}+C\)
\(\displaystyle\int\cos^2x\,\mathrm{d}x\)	\(\cos^2x=\dfrac{1+\cos2x}{2}\)	\(\dfrac{x}{2}+\dfrac{\sin2x}{4}+C\)

Integral	Identity and rewrite	Antiderivative
\(\displaystyle\int\tan^2x\,\mathrm{d}x\)	\(\tan^2x=\sec^2x-1\)	\(\tan x-x+C\)
\(\displaystyle\int\cot^2x\,\mathrm{d}x\)	\(\cot^2x=\mathrm{cosec}^2x-1\)	\(-\cot x-x+C\)

Trigonometric Integrals Using Identities

Trigonometric Integrals Using Identities