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Chapter 4: Problem 41

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(\sec ^{2} \theta+\sec \theta \tan \theta\right) d \theta$$

Short Answer

Expert verified
Question: Find the indefinite integral of the given function: \(\int\left(\sec^2\theta + \sec\theta\tan\theta\right)d\theta\). Answer: \(\tan\theta + \sec\theta + C\) where \(C\) is the integration constant.

Step by step solution

01

Integrate \(\sec^2\theta\)

First, we find the integral for \(\sec^2\theta\). The indefinite integral is given by: $$\int\sec^2\theta d\theta$$ From the standard integration rules, we know that $$\int\sec^2\theta d\theta = \tan\theta + C_1$$ where \(C_1\) is the integration constant.
02

Integrate \(\sec\theta\tan\theta\)

Now, we find the integral for \(\sec\theta\tan\theta\). The indefinite integral is given by: $$\int\sec\theta\tan\theta d\theta$$ Similarly, from the standard integration rules, we know that $$\int\sec\theta\tan\theta d\theta = \sec\theta + C_2$$ where \(C_2\) is another integration constant.
03

Combine the results

Now we combine the results of steps 1 and 2 to obtain the integral of the given integrand: $$\int\left(\sec^2\theta + \sec\theta\tan\theta\right)d\theta = \tan\theta + \sec\theta + C$$ where \(C = C_1 + C_2\) is the overall integration constant.
04

Verify the result by differentiation

To check our work, we differentiate the result we obtained in step 3: $$\frac{d}{d\theta}\left(\tan\theta + \sec\theta + C\right) = \sec^2\theta + \sec\theta\tan\theta$$ Since the derivative matches the original integrand, our solution is correct.

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