Q. 48 Solve the integral鈭�4-xx-2dx... [FREE SOLUTION]

91影视

Chapter 5: Q. 48 (page 465)

Solve the integral
$鈭� \sqrt{4 - x} \sqrt{x - 2} d x$

Short Answer

Expert verified

$\frac{\sin^{-} 1 (x - 3)}{2} + \frac{1}{4} s i n (2 s i n^{-} 1 (x - 3)) + C$

Step by step solution

Step 1. Given

$鈭� \sqrt{4 - x} \sqrt{x - 2} d x$

Step 2. Simplified the integral

$The integral is simplified below . 鈭� \sqrt{4 - x)} \sqrt{(x - 2)} d x = 鈭� \sqrt{- x^{2} + 6 x - 8} d x = 鈭� \sqrt{- (x - 3)^{2} + 1} d x = 鈭� \sqrt{- u^{2} + 1} d x [u = x - 3, d u = d x]$

Step 3. Using the identity

$To solve the integral, le t x = s i n (v) . Now, derivation of the above equation is solved below . u = s i n (v) d u = c o s (v) d v Now, use the identity - s i n^{2} (u) + 1 = c o s^{2} (u) S o, \sqrt{(- u^{2} + 1)} = \sqrt{(- \sin^{2} (v) + 1}) = \sqrt{\cos^{2} (v)} = c o s (y)$

Step 4. Calculation

$The integral is solved below . 鈭� = u 虏 + 1 d x = 鈭� \cos 虏 (v) d v = 鈭� (\frac{1}{2} \cos (2 v) + \frac{1}{2}) d v = 鈭� \frac{1}{2} d v - 鈭� (\frac{1}{2} \cos (2 v) d v) = \frac{v}{2} + \frac{1}{2} 鈭� \frac{1}{2} c o s (w) d w [w = 2 v, d w = 2 d v] = \frac{v}{2} + \frac{1}{4} s i n (w) + C = \frac{v}{2} + \frac{1}{4} s i n (2 v) + C = \frac{\sin^{-} 1 (u)}{2} + \frac{1}{4} s i n (2 s i n^{-} 1 (u)) + C = \frac{\sin^{-} 1 (x - 3)}{2} + \frac{1}{4} s i n (2 s i n^{-} 1 (x - 3)) + C$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve the integral
$鈭� \sqrt{4 - x} \sqrt{x - 2} d x$

Short Answer

Step by step solution

Step 1. Given

Step 2. Simplified the integral

Step 3. Using the identity

Step 4. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Theoretical and Mathematical Physics

Decision Maths

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.