Problem 21

Question

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

Step-by-Step Solution

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Answer

The indefinite integral of $ \frac{\cos t}{1+\sin t} $ with respect to $ t $ is $ \ln |1 + \sin t| $

1Step 1: Substitute variable to simplify fraction

Let's use a substitution, let $ u = 1 + \sin t $. Then, differentiate $ u $ with respect to $ t $ to find $ du $ in terms of $ dt $. We find that $ du = \cos t \, dt $.

2Step 2: Substitute into Integral and Solve

Substituting $ u $ and $ du $ into the original integral gives us $ \int \frac{1}{u} \, du $, which simplifies to $ \ln |u| $.

3Step 3: Back Substitution

To finish the problem, substitute $ u $ back in terms of $ t $. This gives us our final answer, $ \ln |1 + \sin t| $.

Problem 21

Problem 22

Other exercises in this chapter

Problem 21

Evaluate the definite integral of the transcendental function. Use a graphing utility to verify your result. $$ \int_{-\pi / 6}^{\pi / 6} \sec ^{2} x d x $$

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Problem 21

Find the indefinite integral and check the result by differentiation. $$ \int(1-\csc t \cot t) d t $$

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Problem 22

Find the derivative of the function. $h(t)=t-\operatorname{coth} t$

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Problem 22

Evaluate the integral. $$ \int_{0}^{\pi / 2} \frac{\cos x}{1+\sin ^{2} x} d x $$

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