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Chapter 6: Problem 26

Find the integral. $$ \int \frac{\sqrt{1-x}}{\sqrt{x}} d x $$

Short Answer

Expert verified
The integral of \(\frac{\sqrt{1-x}}{\sqrt{x}}\) dx is \(2\ln|\sqrt{x}| - x\)

Step by step solution

01

Apply substitution

Let \( u = \sqrt{x} \). This implies that \( du = (1/2\sqrt{x}) dx \). Now substitute these values into the integral.
02

Simplify the integrand

The integral now takes the form \( \int 2\frac{\sqrt{1-u^2}}{u} du \). This can be split into two separate integrals: \( \int 2\frac{1}{u} du - \int 2u du\)
03

Solve the two integrals

The first integral can be solved as \(2 \ln |u|\) and the second integral as \(-u^2\).
04

Substitute back

Substitute \(u= \sqrt{x}\) back into both solutions to get the final answer: \(2\ln|\sqrt{x}| - x\)

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Most popular questions from this chapter

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=t $$

Determine all values of \(p\) for which the improper integral converges. $$ \int_{0}^{1} \frac{1}{x^{p}} d x $$

Evaluate the definite integral. $$ \int_{0}^{\pi / 4} \tan ^{3} x d x $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of \(f\) is symmetric with respect to the origin or the \(y\) -axis, then \(\int_{0}^{\infty} f(x) d x\) converges if and only if \(\int_{-\infty}^{\infty} f(x) d x\) converges

Consider the integral \(\int_{0}^{3} \frac{10}{x^{2}-2 x} d x\). To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?
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