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Chapter 7: Problem 47

Use the approaches discussed in this section to evaluate the following integrals. $$\int \frac{d x}{x^{1 / 2}+x^{3 / 2}}$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{dx}{x^{1/2}+x^{3/2}}$$. Answer: $$\arctan(x^{1/2})+C$$.

Step by step solution

01

Finding substitution

Let's find the substitution that makes the integral simpler. If we let: $$u = x^{1/2}$$, then we have: $$u^2 = x$$ and $$u^3 = x^{3/2}$$. Consequently, the integral becomes: $$\int \frac{dx}{u+u^3}$$. Moreover, taking the derivative with respect to \(x\), we have: $$\frac{d(u^2)}{dx} = 2u \frac{du}{dx}$$. Solving for \(dx\): $$dx = \frac{2u \ du}{2u^2} = \frac{du}{u}$$. Thus, our integral becomes: $$\int \frac{\frac{du}{u}}{u+u^3} = \int \frac{du}{u(u^2+1)}$$.
02

Partial Fraction Decomposition

Now we will perform partial fraction decomposition on the integrand, and express it as: $$\frac{A}{u} + \frac{Bu+C}{u^2+1}$$. Multiplying both sides by \(u(u^2+1)\), we get: $$1 = A(u^2+1) + (Bu+C)u$$. Comparing the coefficients of the powers of \(u\), we have the following system of linear equations: $$A = 0$$ $$B = 1$$ and $$C = 0$$. Therefore, $$\frac{1}{u(u^2+1)} = \frac{1}{u^2+1}$$.
03

Finding the Antiderivative

Substituting our expression back into the integral, we get: $$\int \frac{du}{u^2+1}$$. This integral can be solved using the relation $$\int \frac{du}{u^2+1} = \arctan(u) +C$$, so our antiderivative in terms of \(u\) is $$\arctan(u) +C$$.
04

Substituting Back with x

Recall that we made the substitution \(u = x^{1/2}\). We need to replace \(u\) with \(x^{1/2}\) to get our final answer. This gives us: $$\arctan(x^{1/2})+C$$. Hence the final solution is: $$\int \frac{dx}{x^{1/2}+x^{3/2}}=\arctan(x^{1/2})+C$$.

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Most popular questions from this chapter

Let \(R\) be the region between the curves \(y=e^{-c x}\) and \(y=-e^{-c x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c \geq 0 .\) The center of mass of \(R\) is located at \((\bar{x}, 0)\) where \(\bar{x}=\frac{\int_{a}^{\infty} x e^{-c x} d x}{\int_{a}^{\infty} e^{-c x} d x} .\) (The profile of the Eiffel Tower is modeled by the two exponential curves.) a. For \(a=0\) and \(c=2,\) sketch the curves that define \(R\) and find the center of mass of \(R\). Indicate the location of the center of mass. b. With \(a=0\) and \(c=2,\) find equations of the lines tangent to the curves at the points corresponding to \(x=0.\) c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any \(a \geq 0\) and any \(c>0 ;\) that is, the tangent lines to the curves \(y=\pm e^{-c x}\) at \(x=a\) intersect at the center of mass of \(R\) (Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique \(332(2004): 571-584 .)\)

Challenge 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 amenabl to partial fractions. Evaluate \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\)

An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-a x^{2}}.\) a. Graph the Gaussian for \(a=0.5,1,\) and 2. b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a). c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{1+e^{x}}$$

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt[4]{x+2}+1} ; x+2=u^{4}$$
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