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Chapter 8: Problem 24

Find the indefinite integral. $$ \int \frac{\cos t}{1+\sin t} d t $$

Short Answer

Expert verified
The integral of \(\frac{\cos t}{1+\sin t} dt\) is \(\ln |1+ \sin t|\)

Step by step solution

01

Identify possible substitutions

Observe that in the integral, the denominator is \(1+ \sin t\), its derivative would not be far from \(\cos t\), which is our numerator. Let \(u = 1 + \sin t\). This would simplify our integral and make our next steps easier.
02

Derive 'du' substitution

Given that \(u = 1+ \sin t\), derive 'du' using the property \(\cos t = \frac{du}{dt}\). So, your 'dt' becomes \(dt = \frac{du}{\cos t}\).
03

Substituting 'u' and 'du'

The integral \(\int \frac{\cos t}{1+\sin t} dt\) becomes \(\int \frac{1}{u} du\). The integral simplifies to a simple logarithmic integral and can be solved easily.
04

Solve the integral

Solving \(\int \frac{1}{u} du\), we get \(\ln |u|\).
05

Re-substitute 'u'

Replace 'u' with the earlier substitution \(u = 1+ \sin t\) which gives the result of our integral as \(\ln |1+ \sin t|\).

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