Question

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{1}^{x} \frac{1}{t} d t $$

Short answer

The integral of the function \( F(x) = \int_{1}^{x} \frac{1}{t} dt \) is \( F(x) = \ln|x| \). The Second Fundamental Theorem of Calculus is confirmed as the derivative of this result is \( \frac{1}{x} \), which is the original function we integrated.

Step-by-step solution

Integration

First, perform the integration for function \(F(x) = \int_{1}^{x} \frac{1}{t} dt\). The integral of \( \frac{1}{t} \) is \( \ln|t| \). Applying the limits of integration from 1 to x, we subtract the value of our antiderivative at x minus the value at 1. This gives us: \( F(x) = \ln|x| - \ln|1| \). Since \( \ln|1| = 0 \), our function simplifies to: \( F(x) = \ln|x| \).

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus states that in the case where \( F(x) = \int_{a}^{x} f(t) dt \), the derivative of \(F\) with respect to \(x\) is \( f(x) \). That is, \( F'(x) = f(x) \). In this case, since the original function was \( \frac{1}{t} \), we expect the derivative of the result from part (a) to be \( \frac{1}{x} \). Computing the derivative of \(F(x) = \ln|x| \) we get \( \frac{1}{x} \), confirming our expectation and demonstrating the theorem.