where C is a constant of integration, then

Engineering

Mathematics

Trigonometric Limits

Elementary Integrals

Integration by Substitution

Question

$If \int \frac{dx}{{(x^{2} - 2 x + 10)}^{2}} = A (\tan^{- 1} (\frac{x - 1}{3}) + \frac{f (x)}{x^{2} - 2 x + 10}) + C$

where C is a constant of integration, then

A = $\frac{1}{54}$ and f(x) = 9(x – 1)²

A = $\frac{1}{81}$ and f (x) = 3(x – 1)

A = $\frac{1}{54}$ and f(x) = 3(x – 1)

A = $\frac{1}{27}$ and f (x) = 9(x – 1)

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$\int \frac{dx}{{(x^{2} - 2 x + 10)}^{2}} = \int \frac{dx}{{({(x - 1)}^{21} + 9)}^{2}}$

Let x – 1 = 3tanθ

dx = 3sec²θdθ

$∴ \frac{1}{27} \int \cos^{2} θdθ = \frac{1}{54} \int (1 + \cos 2 θ) dθ = \frac{1}{54} (θ + \frac{\sin 2 θ}{2})$

$= \frac{1}{54} (\tan^{- 1} (\frac{x - 1}{3}) + \frac{3 (x - 1)}{x^{2} - 2 x + 10}) + C$