Question

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

Short answer

Question: Find the indefinite integral of the given function: $$\int \frac{t+1}{t} dt$$

Step-by-step solution

Break down the integrand into simpler parts

We are given the integral $$\int \frac{t+1}{t} d t$$ To break it down, we can rewrite it as the sum of two separate fractions: $$\int \frac{t}{t} + \frac{1}{t} d t$$ Which simplifies to: $$\int (1 + \frac{1}{t}) dt$$

Integrate each term

Now we can integrate each term of the simplified integrand with respect to t: $$\int 1 dt + \int \frac{1}{t} dt$$ The integral of 1 is simply t, and the integral of \(\frac{1}{t}\) is \(\ln |t|\), which yields: $$t + \ln |t| + C$$, where C is the constant of integration.

Check the result with differentiation

To make sure the solution is correct, we can differentiate our result with respect to t and see if it matches the original integrand. The derivative of t is 1, and the derivative of \(\ln |t|\) is \(\frac{1}{t}\). Combining those, we get: $$\frac{d}{dt} (t + \ln |t| + C) = 1 + \frac{1}{t}$$ Which is the same as the original integrand. This confirms that our indefinite integral is correct. The final answer is: $$\int \frac{t+1}{t} dt = t + \ln |t| + C$$

Key concepts

Integration Techniques

To grasp integration techniques, let's consider how we approach finding the indefinite integral of a function like \(\int \frac{t+1}{t} dt\). The process of integration involves a range of strategies to simplify complex expressions into forms that are easier to integrate.

In our example, the key technique utilized is 'algebraic simplification' where you restructure the integrand into simpler parts before integrating. As seen, the fraction \(\frac{t+1}{t}\) is divided into two separate terms, \(\int \left(1 + \frac{1}{t}\right) dt\), making it easier to apply known integration rules. Each term is then integrated individually; \(\int 1 dt\) is straightforward, while \(\int \frac{1}{t} dt\) requires recognizing that it corresponds to the natural logarithm function. The combination of these integrated terms then gives us the complete solution. Recognizing how to decompose an integrand and apply rules and identities is fundamental in mastering integration techniques.

Natural Logarithm

In the context of integration, the natural logarithm often appears when dealing with terms of the form \(\int \frac{1}{x} dx\). This specific integral is fundamental and leads to the natural logarithm \(\ln |x|\).

The natural logarithm, denoted as \(\ln\), is the inverse function to the exponential function with base \(e\), where \(e\) is approximately equal to 2.71828. It is important to note that \(\ln\) is only defined for positive numbers; hence we use the absolute value sign around the variable to ensure the argument of \(\ln\) is always positive, as demonstrated in \(\ln |t|\).

Due to these properties, the natural logarithm plays a crucial role in integration, particularly when reverse engineering the differentiation of \(\ln\) since the derivative of \(\ln |x|\) gives us \(\frac{1}{x}\), helping to confirm the correctness of an integration process involving \(\frac{1}{x}\) terms.

Constant of Integration

When we perform indefinite integration, we're finding the most general form of the antiderivative for a function. This introduces the concept of the 'constant of integration', denoted by \(C\).

Every time you compute an indefinite integral, the result includes a \(C\) because differentiation of a constant yields zero, leaving us with many possible antiderivatives that differ by a constant amount. This concept is seen after integrating \(\int \frac{t+1}{t} dt\), resulting in \(t + \ln |t| + C\).

While definite integrals compute precise values over a given interval, the indefinite integral represents a family of functions that all satisfy the original function when differentiated. The inclusion of \(C\) in our solution acknowledges all possible antiderivatives. Confirming that differentiation of the integrated function \(t + \ln |t| + C\) returns the original integrand validates both the process of integration and the necessity of including the constant of integration.