Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Evaluate the Equation\(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {co{t^3}x\,dx} \)

The answer is \(\frac{1}{2} - \frac{{ln\left( 2 \right)}}{2}\)

First, convert \(co{t^3}x\) in \(cosec\,x\) using trigonometric identities and then substituting \(cosec\,x\) with \(u\), solve the definite integral.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Converting \(co{t^3}x\) in \(cosec\,x\) using the trigonometric identity.

\(\begin{aligned}{l}\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {co{t^3}x\,dx} \\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {cot\,x\left( {co{t^2}x} \right)\,dx} \\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {cot\,x\left( {cose{c^2}x - 1} \right)\,dx} \end{aligned}\)

Step 2: Putting \(cosec\,x = u\)

Let \(cosec\,x = u\)

\(\begin{aligned}{l} \Rightarrow - cot x cosec x dx &= du\\ \Rightarrow cot x dx &= - \frac{{du}}{{cosec\,x}}\\ \Rightarrow cotx\,dx &= - \frac{{du}}{u}\\ \Rightarrow - \frac{{du}}{u} &= cot\,x\,dx\,\,.....(1)\end{aligned}\)

So,

\(\begin{aligned}{l} &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {cot\,x\left( {cose{c^2}x - 1} \right)\,dx} \\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\left( {cose{c^2}x - 1} \right)cot\,x\,dx} \\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\left( {{u^2} - 1} \right)\left( { - \frac{{du}}{u}} \right)\,\,\,\,\left( {From\,\,(i)} \right)} \\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\frac{{ - \left( {{u^2} - 1} \right)}}{u}} \,du\\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\left( { - u + \frac{1}{u}} \right)} \,du\\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\left( {\frac{1}{u} - u} \right)} \,du\end{aligned}\)

Step 3: Applying linearity and solving the integral.

\(\begin{aligned}{l} &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\left( {\frac{1}{u} - u} \right)} \,du\\ &= \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\left( {\frac{1}{u}} \right)} \,du - \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {u\,} du\\ &= \left( {ln\left| u \right|} \right)_{\frac{\pi }{4}}^{\frac{\pi }{2}} - \left( {\frac{{{u^2}}}{2}} \right)_{\frac{\pi }{4}}^{\frac{\pi }{2}}\end{aligned}\)

Step 4: Putting back \(u = cosec\,x\) and putting the limit.

\( = \left( {ln\left| u \right|} \right)_{\frac{\pi }{4}}^{\frac{\pi }{2}} - \left( {\frac{{{u^2}}}{2}} \right)_{\frac{\pi }{4}}^{\frac{\pi }{2}}\)

\( = \left( {ln\left| {cosec\,x} \right|} \right)_{\frac{\pi }{4}}^{\frac{\pi }{2}} - \left( {\frac{{cose{c^2}x}}{2}} \right)_{\frac{\pi }{4}}^{\frac{\pi }{2}}\)

\( = \left( {ln\left| {cosec\,\left( {\frac{\pi }{2}} \right)} \right| - ln\left| {cosec\,\left( {\frac{\pi }{4}} \right)} \right|} \right) - \left( {\frac{{cose{c^2}\left( {\frac{\pi }{2}} \right)}}{2} - \frac{{cose{c^2}\left( {\frac{\pi }{4}} \right)}}{2}} \right)\)

\( = \left( {ln\left| 1 \right| - ln\left| {\sqrt 2 } \right|} \right) - \left( {\frac{{{1^2}}}{2} - \frac{{{{\left( {\sqrt 2 } \right)}^2}}}{2}} \right)\)

\( = 0 - ln\left| {\sqrt 2 } \right| - \left( {\frac{1}{2} - \frac{2}{2}} \right)\)

\( = - ln\left| {\sqrt 2 } \right| - \left( { - \frac{1}{2}} \right)\)

\( = - ln{\left( 2 \right)^{\frac{1}{2}}} + \frac{1}{2}\)

\( = \frac{1}{2} - ln{\left( 2 \right)^{\frac{1}{2}}}\)

\( = \frac{1}{2} - \frac{{ln\left( 2 \right)}}{2}\)

Hence, \(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {co{t^3}x\,dx} = \frac{1}{2} - \frac{{ln\left( 2 \right)}}{2}\)\(\)

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Essential Calculus: Early Transcendentals

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Short Answer

Evaluate the Equation\(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {co{t^3}x\,dx} \)

Step by Step Solution

Step 1: Converting \(co{t^3}x\) in \(cosec\,x\) using the trigonometric identity.

Step 2: Putting \(cosec\,x = u\)

Step 3: Applying linearity and solving the integral.

Step 4: Putting back \(u = cosec\,x\) and putting the limit.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the Equation\(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {co{t^3}x\,dx} \)

Step by Step Solution

Step 1: Converting \(co{t^3}x\) in \(cosec\,x\) using the trigonometric identity.

Step 2: Putting \(cosec\,x = u\)

Step 3: Applying linearity and solving the integral.

Step 4: Putting back \(u = cosec\,x\) and putting the limit.

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