/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 43

Evaluate the following integrals or state that they diverge. $$\int_{0}^{1} \frac{x^{3}}{x^{4}-1} d x$$

Short Answer

Expert verified
Based on the step-by-step solution provided above, the final answer to the integral is that it diverges. This is due to the natural logarithm of zero being undefined when evaluating the integral, causing the expression to become impossible to compute.

Step by step solution

01

Substitution

Let \(u = x^4 - 1\), then we can find \(du\) by differentiation. We get \(du = 4x^3 dx\). Now, we can rewrite the integral as: $$\int_{0}^{1} \frac{x^{3}}{x^{4}-1} d x = \frac{1}{4} \int_{-1}^{0} \frac{1}{u} du$$ We changed the limits of integration according to the substitution: \(u(0) = -1\) and \(u(1)=0\).
02

Evaluate the integral

The integral now becomes: $$\frac{1}{4} \int_{-1}^{0} \frac{1}{u} du$$ This is an integral of the form \(\int \frac{1}{u} du\), which is equal to the natural logarithm function. Therefore, we have: $$\frac{1}{4} [\ln(|u|)]_{-1}^{0}$$
03

Use the fundamental theorem of calculus

Now, we just need to plug in the limits of integration into the natural logarithm function and subtract, following the fundamental theorem of calculus: $$\frac{1}{4} [\ln(|0|)-\ln(|-1|)]$$ However, when trying to evaluate \(\ln(|0|)\), we get undefined value as the natural logarithm of zero is not possible. Therefore, the integral diverges.

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