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Essential Calculus: Early Transcendentals
Found in: Page 363
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Evaluate the integral\(\int {\frac{{{\rm{sin(lnt)}}}}{{\rm{t}}}} {\rm{dt}}\)

Evaluation’s successor is \(\frac{{{\rm{sin(lnt)}}}}{{\rm{t}}}{\rm{dt = - cos(lnt) + C}}\)

See the step by step solution

Step by Step Solution

Step1: Integral differential.

\(\begin{aligned}{l}\frac{{{\rm{sin(lnt)}}}}{{\rm{t}}}{\rm{dt}}\\{\rm{ = }}\;\;\;{\rm{sin(lnt)}}\frac{{\rm{1}}}{{\rm{t}}}{\rm{dt}}\end{aligned}\)

On differentiating, we'll use a substitute.

\(\begin{aligned}{l}{\rm{ = sin(u)du}}\\{\rm{ = - cos(u) + C}}\end{aligned}\)

Step2: Restore the original position\({\rm{u = lnt}}\).

\({\rm{ = - cos(lnt) + C}}\)

Finally, the original integral position is\(\frac{{{\rm{sin(lnt)}}}}{{\rm{t}}}{\rm{dt = - cos(lnt) + C}}\).

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