Answered: Find ∫ dx/ x√(x−1) .

Name: Find ∫ dx/ x√(x−1) .
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: Find ∫ dx/ x√(x−1) .

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.3: Logarithmic Functions

Problem 61SE: Is there a number x such that ln x=2 ? If so, whatis that number? Verify the result.

See similar textbooks

Related questions

Topic Video

Question

Find ∫ dx/ x√(x−1) .

Expert Solution

Step 1

To evaluate:

$\int \frac{1}{x \sqrt{x - 1}} d x$

Step 2

Consider, $\int \frac{1}{x \sqrt{x - 1}} d x$

Substitute, $u = \sqrt{x - 1}$ . . .(1)

$\Rightarrow$ $d u = \frac{1}{2 \sqrt{x - 1}} d x$

From (1)

$\begin{array}{rcl} u^{2} & = & x - 1 \\ x & = & 1 + u^{2} \end{array}$

Thus, the given integration becomes

$\begin{array}{rcl} \int \frac{1}{x \sqrt{x - 1}} d x & = & \int \frac{2}{1 + u^{2}} d u \\ = & 2 \int \frac{1}{1 + u^{2}} d u \\ = & 2 a r c \tan (u) + C . . . Using \int \frac{1}{1 + x^{2}} d x = a r c \tan (x) + C \end{array}$

Back substitute $u = \sqrt{x - 1}$ , to get

$\int \frac{1}{x \sqrt{x - 1}} d x = 2 a r c \tan (\sqrt{x - 1}) + C$

Step by step

Solved in 3 steps

Knowledge Booster

$Chain Rule$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.