Chapter 5: Q. 48 (page 465)

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

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

$\int \sqrt{4 - x} \sqrt{x - 2} d x$

Step 2. Simplified the integral

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

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

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

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

Short Answer

Step by step solution

Step 1. Given

Step 2. Simplified the integral

Step 3. Using the identity

Step 4. Calculation

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Solve the integral∫4-xx-2dx

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

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Statistics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

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