Name: Integral of Irrational Function: (ax^2+bx+c)^1/2 dx
Uploaded: 2019-07-19
Duration: 8 min 38 s
Description: Let's see how to integrate when a complete surd is present in the numerator. Check this video out since it uses an example to teach a complex concept.

Question

Evaluate:

\int \sqrt{\frac{5 - x}{x - 2}} d x on (2, 5)

.

Accepted Answer

Let $I = \int \sqrt{\frac{5 - x}{x - 2}} d x$

$\Rightarrow I = \int \sqrt{\frac{5 - x}{x - 2} \times \frac{5 - x}{5 - x}} d x$

$\Rightarrow I = \int \frac{5 - x}{\sqrt{5 x - x^{2} - 10 + 2 x}} d x = \int \frac{5 - x}{\sqrt{7 x - x^{2} - 10}} d x$

We know that, $\int \frac{p x + q}{\sqrt{a x^{2} + b x + c}} d x = p x + q = A \frac{d}{d x} (a x^{2} + b x + c) + B$

$∴ \int \frac{5 - x}{\sqrt{7 x - x^{2} - 10}} d x = 5 - x = A \frac{d}{d x} (7 x - x^{2} - 10) + B$

$\Rightarrow 5 - x = A (7 - 2 x) + B$

On equating the coefficients and constants, we get

$A = \frac{1}{2}, B = \frac{3}{2}$

$∴ 5 - x = \frac{1}{2} (7 - 2 x) + \frac{3}{2}$

$∴ I = \int \frac{5 - x}{\sqrt{7 x - x^{2} + 0}} d x = \frac{1}{2} \int \frac{7 - 2 x}{\sqrt{7 x - x^{2} - 10}} d x + \frac{3}{2} \int \frac{1}{\sqrt{7 x - x^{2} - 10}} d x$

We know that $\int \frac{f^{'} (x)}{\sqrt{f (x)}} d x = 2 \sqrt{f (x)}$ and consider

$I_{1} = \int \frac{1}{\sqrt{7 x - x^{2} - 10}} d x = \int \frac{d x}{\sqrt{- (x^{2} - 7 x + 10)}}$

$= \int \frac{d x}{\sqrt{- (x^{2} - 7 x + {(\frac{7}{2})}^{2} - {(\frac{7}{2})}^{2} + 10)}} = \int \frac{d x}{\sqrt{- {(x - \frac{7}{2})}^{2} + {(\frac{3}{2})}^{2}}}$

$\Rightarrow I_{1} = \int \frac{d x}{\sqrt{{(\frac{3}{2})}^{2} - {(x - \frac{7}{2})}^{2}}} = \sin^{- 1} \frac{(x - \frac{7}{2})}{(\frac{3}{2})} + c$

Hence, $I = \sqrt{7 x - x^{2} + 10} + \frac{3}{2} \sin^{- 1} [\frac{2 x - 7}{3}] + C$

Evaluate: $\int \sqrt{\frac{5 - x}{x - 2}} d x on (2, 5)$ .

Important Questions on Integration

Learn and Prepare for any exam you want !

Evaluate: ∫5-xx-2dx on (2,5).

Important Questions on Integration

Learn and Prepare for any exam you want !

Evaluate: $\int \sqrt{\frac{5 - x}{x - 2}} d x on (2, 5)$ .