/*! 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 17 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 17

Evaluate the following integrals. $$\int_{1}^{e^{2}} \frac{\ln ^{2}\left(x^{2}\right)}{x} d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the function $$\frac{\ln^2\left(x^2\right)}{x}$$ from $x=1$ to $x=e^2$. Answer: $$\int_{1}^{e^2} \frac{\ln^2\left(x^2\right)}{x} dx = \frac{32}{3}$$.

Step by step solution

01

Simplify the logarithm function

Using the logarithm property $$\ln(a^b)=b\ln(a)$$, we can simplify the given integrand: $$ \frac{\ln^2\left(x^2\right)}{x} = \frac{(2\ln(x))^2}{x} $$ which further simplifies to: $$ \frac{4 \ln^2(x)}{x} $$
02

Integrate the simplified function

Now, we need to evaluate the integral: $$ \int_{1}^{e^2} \frac{4 \ln^2(x)}{x} dx $$ Let's use $$u$$-substitution: $$ u = \ln(x) \Rightarrow x = e^u $$ and find the derivative: $$ d x = e^u d u $$ Now, we also need to rewrite the limits of integration in terms of $$u$$. For the lower limit, when $$x = 1$$: $$ u = \ln(1) = 0 $$ For the upper limit, when $$x = e^2$$: $$ u = \ln(e^2) = 2 $$ The integral becomes: $$ \int_{0}^{2} \frac{4 u^2}{e^u} e^u d u = \int_{0}^{2} 4 u^2 d u $$
03

Evaluate the integral

We can now solve the integral: $$ \int_{0}^{2} 4 u^2 d u $$ By applying the power rule: $$ \int 4 u^2 du = 4 \int u^2 du = 4 \left(\frac{u^3}{3}\right) $$ Now, we need to apply the limits of integration: $$ 4 \left(\frac{u^3}{3}\right)\Big|_{0}^{2} = 4 \left(\frac{2^3}{3} - \frac{0^3}{3}\right) = 4 \left(\frac{8}{3}\right) = \frac{32}{3} $$ The integral of the original function is: $$ \int_{1}^{e^2} \frac{\ln^2\left(x^2\right)}{x} dx = \frac{32}{3} $$

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Most popular questions from this chapter

On the interval \([0,2],\) the graphs of \(f(x)=x^{2} / 3\) and \(g(x)=x^{2}\left(9-x^{2}\right)^{-1 / 2}\) have similar shapes. a. Find the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval [0,2] b. Find the area of the region bounded by the graph of \(g\) and the \(x\) -axis on the interval [0,2] c. Which region has the greater area?

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int \frac{d x}{1-\cos x}$$

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Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{x-\sqrt[4]{x}} ; x=u^{4}$$

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