<p class="paragraph-header">Changing it to an integrable form such as </p>
<ol class="i-list"><li><div class="list-item-paragraph">\(\sin nx\), \(\cos nx\), \({{\sec }^{2}}nx\) and \({{\operatorname{cosec}}^{2}}nx\).</div></li><li><div class="list-item-paragraph">\(\tan ax\), \(\cot ax\), \(\operatorname{cosec}ax\), \(\sec ax\) from MF26. </div></li></ol>

Use trigonometric identities to rewrite the integrand as a standard or recognisable form.

\(\int\sin(nx)\,\mathrm dx=-\frac1n\cos(nx)+C,\qquad\int\cos(nx)\,\mathrm dx=\frac1n\sin(nx)+C\)

\(\int\sec^2(nx)\,\mathrm dx=\frac1n\tan(nx)+C,\qquad\int\operatorname{cosec}^2(nx)\,\mathrm dx=-\frac1n\cot(nx)+C\)

\(\int\tan(ax)\,\mathrm dx=-\frac1a\ln|\cos(ax)|+C,\qquad\int\cot(ax)\,\mathrm dx=\frac1a\ln|\sin(ax)|+C\)

\(\int\sec(ax)\,\mathrm dx=\frac1a\ln|\sec(ax)+\tan(ax)|+C\)

\(\int\operatorname{cosec}(ax)\,\mathrm dx=\frac1a\ln|\operatorname{cosec}(ax)-\cot(ax)|+C\)

Source	Standard forms
Secondary level	\(\sin x,\ \cos x,\ \sec^2x,\ \mathrm{cosec}^2x\)
MF27	\(\tan x,\ \cot x,\ \mathrm{cosec}\,x,\ \sec x\)

Standard Trigonometric Integrals

Standard Trigonometric Integrals

Video Solution:

Solution:

Solution locked