Question

Evaluate the Equation \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {co{t^2}xdx} \)

Short answer

First, convert \(co{t^2}x\) in terms of \(cosec x\) using trigonometric identities and solve the definite integral. Therefore , \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {co{t^2}xdx} = \sqrt 3 - \frac{\pi }{3}\).

Step-by-step solution

Converting \(co{t^2}x\) in \(cosec\,x\) using the trigonometric identity.

=\(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {co{t^2}xdx} \)

\( = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\left( {cose{c^2}x - 1} \right)dx} \)

Applying linearity and solving the integral.

\( = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\left( {cose{c^2}x - 1} \right)dx} \)

\( = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {cose{c^2}x\,dx} - \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {1\,dx} \)

\( = \left( { - cot\,x} \right)_{\frac{\pi }{6}}^{\frac{\pi }{2}} - \left( x \right)_{\frac{\pi }{6}}^{\frac{\pi }{2}}\)

Putting the limit.

\( = \left( { - cot\,\left( {\frac{\pi }{2}} \right) - \left( { - cot\left( {\frac{\pi }{6}} \right)} \right)} \right) - \left( {\frac{\pi }{2} - \frac{\pi }{6}} \right)\)

\( = \left( { - 0 - \left( {\sqrt 3 } \right)} \right) - \left( {\frac{{3\pi - \pi }}{6}} \right)\)

\( = \sqrt 3 - \frac{{2\pi }}{6}\)

\( = \sqrt 3 - \frac{\pi }{3}\)

Hence, \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{2}} {co{t^2}xdx} = \sqrt 3 - \frac{\pi }{3}\).

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