Chapter 5: Q. 47 (page 407)

Solve the integral
$\int \sqrt{3 - x} \sqrt{x - 1} d x$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given

The given function is $\int \sqrt{3 - x} \sqrt{x - 1} d x$

Step 2. Simplified the integral

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

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

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

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Solve the integral
$\int \sqrt{3 - x} \sqrt{x - 1} d x$

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∫3-xx-1dx

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Step by step solution

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One App. One Place for Learning.

Most popular questions from this chapter

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Solve the integral
$\int \sqrt{3 - x} \sqrt{x - 1} d x$