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Chapter 4: Problem 55

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{t+1}{t} d t$$

Short Answer

Expert verified
Question: Determine the indefinite integral of \(\frac{t+1}{t}\) and verify the result by differentiation. Answer: \(\int \frac{t+1}{t} dt = t + ln|t| + C\)

Step by step solution

01

Rewrite the integrand

Let's rewrite the function inside the integral in a more convenient form that we can integrate easily: $$\int \frac{t+1}{t} dt = \int \frac{t}{t} + \frac{1}{t} dt = \int 1 + \frac{1}{t} dt$$
02

Integrating the rewritten function

Now, we can integrate the rewritten function term by term: $$\int 1 + \frac{1}{t} dt = \int 1 dt + \int \frac{1}{t} dt$$ Recalling the anti-derivatives of these simpler functions: $$\int 1 dt = t + C_1$$ $$\int \frac{1}{t} dt = ln|t| + C_2$$
03

Combining the results of integration

Combining the results from Step 2, we get the final integral: $$\int 1 + \frac{1}{t} dt = t + ln|t| + C$$ The combined constant is given by \(C = C_1 + C_2\).
04

Checking the result by differentiation

To confirm our result is correct, we need to differentiate it and check if it gives us the original function. Calculating the derivative of \(t + ln|t|\): $$\frac{d}{dt} (t + ln|t|) = 1 + \frac{1}{t}$$ So our indefinite integral is correct because the result of differentiating it gives us back the original function: $$\int \frac{t+1}{t} dt = t + ln|t| + C$$

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