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Chapter 7: Problem 25

Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} \frac{3 x^{2}+1}{x^{3}+x} d x$$

Short Answer

Expert verified
Question: Determine whether the integral converges or diverges: $$\int_{1}^{\infty} \frac{3x^2 +1}{x^3 +x} dx$$ Answer: The given integral diverges.

Step by step solution

01

Write the integral as the limit of a proper integral

We can rewrite the given integral as the limit of a proper integral: $$\lim_{b\to\infty}\int_{1}^{b} \frac{3x^2 +1}{x^3 +x} dx$$ Our focus is now to evaluate this limit and check its convergence.
02

Perform Integration by Substitution

We recognize the expression in the denominator (\(x^3+x\)) is the derivative of \(\frac{1}{4}x^4+\frac{1}{2}x^2\) (up to a constant factor). Let's perform a substitution: Let $$u = x^3+x$$ Then, $$\frac{du}{dx} = 3x^2+1$$ And the differential, $$dx = \frac{du}{3x^2+1}$$ We need to change the limits of integration, as follows: When \(x=1\), \(u=1+1\) which gives \(u=2\) When \(x=b\), \(u=b^3+b\) which gives \(u=b(b^2+1)\) Now substitute and change the limits of integration: $$\lim_{b\to\infty}\int_1^b \frac{3x^2+1}{x^3+x} dx = \lim_{b\to\infty}\int_2^{b(b^2+1)} \frac{1}{u} du$$
03

Evaluate the limit of the proper integral

First, we evaluate the integral: $$\int_2^{b(b^2+1)}\frac{1}{u}du = [\ln(u)]_2^{b(b^2+1)}$$ Now, we find the limit as \(b\to\infty\): $$\lim_{b\to\infty}\left[\ln(b(b^2+1)) - \ln(2)\right]$$ As \(b\to\infty\), the first term inside the bracket, \(\ln(b(b^2+1))\) goes to \(\infty\). Therefore, the limit does not exist, and the integral diverges.
04

Final answer

Since the limit diverges, we can state that the integral diverges: $$\int_{1}^{\infty} \frac{3x^2 +1}{x^3 +x} dx \quad \text{diverges}$$

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Most popular questions from this chapter

Show that with the change of variables \(u=\sqrt{\tan x}\) the integral \(\int \sqrt{\tan x} d x\) can be converted to an integral amenable to partial fractions. Evaluate \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\)

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \sec ^{4} 4 t d t$$

Approximate the following integrals using Simpson's Rule. Experiment with values of \(n\) to ensure that the error is less than \(10^{-3}\). \(\int_{0}^{\pi} \sin 6 x \cos 3 x d x=\frac{4}{9}\)

Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(7 .\) Make \(a\) table similar to Table 7.8 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{0}^{4}\left(3 x^{5}-8 x^{3}\right) d x=1536\)

Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(7 .\) Make \(a\) table similar to Table 7.8 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{0}^{\pi} e^{-t} \sin t d t=\frac{1}{2}\left(e^{-\pi}+1\right)\)
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