Question

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) The Fundamental Theorem of Calculus applies to \(\int_0^2\frac{1}{1-x}dx\).

(b) The Fundamental Theorem of Calculus applies to \(\int_2^{\infty}\frac{1}{1-x}dx\).

(c) If \(f(x)\) is positive and decreasing, then the area between the graph of \(f(x)\) and the \(x-\)axis on \([1,\infty)\) must be finite.

(d) If \(f(x)\) is positive and decreasing, then the area between the graph of \(f(x)\) and the \(x-\)axis on the \([1,\infty)\) must be infinite.

(e) If \(p>1\), then \(\int_1^{\infty}\frac{1}{x^p}dx=\frac{1}{1-p}\).

(f) If \(p<1\), then \(\int_0^{1}\frac{1}{x^p}dx=\frac{1}{1-p}\).

(g) If \(\int_1^{\infty} f(x)dx\) converges and \(0\leq f(x)\leq g(x)\) for all \(x\in[1,\infty)\), then \(\int_1^{\infty}g(x)dx\) must diverge.

(h) If \(\int_1^{\infty} f(x)dx\) diverges and \(0\leq f(x)\leq g(x)\) for all \(x\in[1,\infty)\), then \(\int_1^{\infty}g(x)dx\) must diverge.

Short answer

(a) True

(b)

Step-by-step solution

Part (a) Step 1: State true or false

The Fundamental Theorem of Calculus applies to \(\int_0^2\frac{1}{1-x}dx\).

Here the limit is finite so the theorem is applicable.

It is a true statement.