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Chapter 6: Problem 6

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{3}^{4} \frac{1}{(x-3)^{3 / 2}} d x $$

Short Answer

Expert verified
The given improper integral is convergent and its value is -4.

Step by step solution

01

Identify Type of Improper Integral

The integral is of the type \( \int_{a}^{b} f(x) dx \) where f(x) is not continuous on the interval [a, b]. The point of discontinuity exists at x = 3 due to the zero denominator.
02

Rewrite the Integral as a Limit Expression

The improper integral can be rewritten as a limit expression. Here it can be rewritten as \( \lim_{{t \to 3^{+}}} \int_{t}^{4} \frac{1}{(x-3)^{3 / 2}} dx. \)
03

Evaluate the Integral

The integral can be evaluated by substitution where let \( u = x - 3 \), then the du will replace dx and then by powers rule, evaluate the antiderivative. Substitution gives: \( \lim_{{t \to 3^{+}}} -2 \int_{t}^{4} u^{-1/2} du \).By powers rule, this will result in: \( \lim_{{t \to 3^{+}}} [ -2 * 2 * u^{1/2} ]_{t}^{4} . \)
04

Evaluate the Limit

Substitute t and 4 in the limit and evaluate it. \( \lim_{{t \to 3^{+}}} (-4 \sqrt{1} + 4 \sqrt{t-3} ) . \) As t approaches to 3+, \(t-3\) approaches 0 and the limit evaluates to \( -4 \).

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Most popular questions from this chapter

Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=0\).

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \sec ^{5} \pi x \tan \pi x d x $$

Define the terms converges and diverges when working with improper integrals.

Find the values of \(a\) and \(b\) such that \(\lim _{x \rightarrow 0} \frac{a-\cos b x}{x^{2}}=2\).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sinh a t $$
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