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Chapter 4: Problem 12

Find the indefinite integral. $$ \int \frac{1}{x^{2 / 3}\left(1+x^{1 / 3}\right)} d x $$

Short Answer

Expert verified
The integral of \( \int \frac{1}{x^{2 / 3}(1+x^{1 / 3})} dx \) is \( 3 \ln |1+ x^{1 / 3}|+ C \)

Step by step solution

01

Identify Substitution

Let's take \( u = x^{1 / 3} \). Then, \(\frac{du}{dx} = \frac{1}{3}x^{-2 / 3} \). Then, \( dx = 3u^2 du \). This will simplify the denominator and make our integral easier to solve.
02

Substitute into Integral

Now, substitute in the variable u and the new dx into the original integral: \(\int \frac{1}{u^2 (1+u)} 3u^2 du \). This simplifies to \(3 \int \frac{1}{1+u} du \).
03

Integrate

Now, we can easily compute this integral, which results to \(3 \ln |1+u| + C \). This is the result in terms of u.
04

Substitute Back

In the last step, we need to substitute back to our original variable x. Since we have set \( u = x^{1 / 3} \), the final result is hence \(3 \ln |1+ x^{1 / 3}|+ C \).

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Most popular questions from this chapter

A function \(f\) is defined below. Use geometric formulas to find \(\int_{0}^{8} f(x) d x\) $$f(x)=\left\\{\begin{array}{ll}4, & x<4 \\ x, & x \geq 4\end{array}\right.$$

Use the error formulas in Theorem 4.19 to estimate the error in approximating the integral, with \(n=4\), using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} \sin (\pi x) d x $$

(a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point.(b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). $$ \frac{d y}{d x}=x^{2}-1, \quad(-1,3) $$

Use the table of values to find lower and upper estimates of $$\int_{0}^{10} f(x) d x$$ Assume that \(f\) is a decreasing function. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline f(x) & 32 & 24 & 12 & -4 & -20 & -36 \\ \hline \end{array} $$

The table lists several measurements gathered in an experiment to approximate an unknown continuous function \(y=f(x)\). (a) Approximate the integral \(\int_{0}^{2} f(x) d x\) using the Trapezoidal Rule and Simpson's Rule. \begin{tabular}{|c|c|c|c|c|c|} \hline\(x\) & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 \\ \hline\(y\) & 4.32 & 4.36 & 4.58 & 5.79 & 6.14 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\(x\) & 1.25 & 1.50 & 1.75 & 2.00 \\ \hline\(y\) & 7.25 & 7.64 & 8.08 & 8.14 \\ \hline \end{tabular} (b) Use a graphing utility to find a model of the form \(y=a x^{3}+b x^{2}+c x+d\) for the data. Integrate the resulting polynomial over [0,2] and compare your result with your results in part (a).
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