Chapter 17: Problem 5

Find $\int \frac{\mathrm{d} x}{(1-x) \sqrt{x}}$.

Short Answer

Expert verified

Question: Evaluate the integral $\int \frac{\mathrm{d} x}{(1-x) \sqrt{x}}$. Answer: $\int \frac{\mathrm{d}x}{(1-x) \sqrt{x}} = \frac{1}{2}\ln\left|\frac{1+\sqrt{x}}{1-\sqrt{x}}\right| + C$

Step by step solution

Choose an appropriate substitution

Let's make a substitution to simplify the integrand. We will choose $u = \sqrt{x}$, so $x = u^2$. To find the differential, we take the derivative of $x$ with respect to $u$ and get $\mathrm{d}x = 2u\,\mathrm{d}u$.

Substitute and simplify integrand

Now, let's substitute the expressions for $x$ and $\mathrm{d}x$ into the original integral: $$\int \frac{\mathrm{d}x}{(1-x)\sqrt{x}} = \int \frac{2u\,\mathrm{d}u}{(1-u^2)u}.$$ Simplifying the integrand, we get: $$\int \frac{2u\,\mathrm{d}u}{(1-u^2)u} = 2\int \frac{\mathrm{d}u}{1-u^2}.$$

Recognize a known integral

The simplified integral is now in the form of a known integration result: $$\int \frac{\mathrm{d}u}{1-u^2}.$$ This integral can be found using the definition of the natural logarithm function and the properties of hyperbolic functions. The result is: $$\int \frac{\mathrm{d}u}{1-u^2} = \frac{1}{2}\ln\left|\frac{1+u}{1-u}\right| + C.$$

Substitute back for x

Finally, let's substitute back in terms of $x$ using the original substitution $u = \sqrt{x}$: $$\frac{1}{2}\ln\left|\frac{1+\sqrt{x}}{1-\sqrt{x}}\right| + C.$$ So, the antiderivative of the given integral is: $$\int \frac{\mathrm{d}x}{(1-x) \sqrt{x}} = \frac{1}{2}\ln\left|\frac{1+\sqrt{x}}{1-\sqrt{x}}\right| + C.$$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find \(\int \frac{\mathrm{d} x}{(1-x) \sqrt{x}}\).

Short Answer

Step by step solution

Choose an appropriate substitution

Substitute and simplify integrand

Recognize a known integral

Substitute back for x

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Medical Physics

Classical Mechanics

Mechanics and Materials

Force

Measurements

Study anywhere. Anytime. Across all devices.