1-calculste-f-a-0-dx-x-2-x-a-with-a-gt-1-2-calculate-f-a-at-form-of-integral-then-find-its-value-

Question Number 77755 by abdomathmax last updated on 09/Jan/20

1) c a l c u l s t e f (a) = \int_{0}^{\infty} \frac{d x}{\sqrt{x^{2} - x + a}} w i t h a > 1

2) c a l c u l a t e f^{^{'}} (a) a t f o r m o f i n t e g r a l t h e n f i n d

i t s v a l u e .

Commented by mathmax by abdo last updated on 11/Jan/20

f o r g i v e f (a) = \int_{0}^{1} \frac{d x}{\sqrt{x^{2} - x + a}} w i t h a > 1

Commented by mathmax by abdo last updated on 11/Jan/20

f (a) = \int_{0}^{1} \frac{d x}{\sqrt{x^{2} - x + a}} w e h a v e x^{2} - x + a = x^{2} - 2 \frac{x}{2} + \frac{1}{4} + a - \frac{1}{4}

= {(x - \frac{1}{2})}^{2} + \frac{4 a - 1}{4} a n d \frac{4 a - 1}{4} > 0 f h a n g e m e n t x - \frac{1}{2} = \frac{\sqrt{4 a - 1}}{2} u

g i v e f (a) = \int_{- \frac{1}{\sqrt{4 a - 1}}}^{\frac{1}{\sqrt{4 a - 1}}} \frac{1}{\frac{\sqrt{4 a - 1}}{2} (\sqrt{1 + u^{2}}} \times \frac{\sqrt{4 a - 1}}{2} d u

= \int_{\frac{- 1}{\sqrt{4 a - 1}}}^{\frac{1}{\sqrt{4 a - 1}}} \frac{d u}{\sqrt{1 + u^{2}}} = 2 \int_{0}^{\frac{1}{\sqrt{4 a - 1}}} \frac{d u}{\sqrt{1 + u^{2}}} = 2 {[l n (u + \sqrt{= u^{2} + 1})]}_{0}^{\frac{1}{\sqrt{4 a - 1}}}

= 2 {l n (\frac{1}{\sqrt{4 a - 1}} + \sqrt{\frac{1}{4 a - 1} + 1})} = 2 l n (\frac{1}{\sqrt{4 a - 1}} + \frac{2 \sqrt{a}}{\sqrt{4 a - 1}})

= 2 l n (\frac{2 \sqrt{a} + 1}{\sqrt{4 a - 1}}) = 2 l n (2 \sqrt{a} + 1) - l n (4 a - 1)

\Rightarrow f (a) = 2 l n (2 \sqrt{a} + 1) - l n (4 a - 1)

2) w e h a v e f^{^{'}} (a) = \int_{0}^{1} \frac{\partial}{\partial a} (\frac{1}{\sqrt{x^{2} - x + a}}) d x

= \int_{0}^{1} \frac{\partial}{\partial a} {{(x^{2} - x + a)}^{- \frac{1}{2}}} d x = \int_{0}^{1} - \frac{1}{2} {(x^{2} - x + a)}^{- \frac{3}{2}} d x

= - \frac{1}{2} \int_{0}^{1} \frac{d x}{(x^{2} - x + a) \sqrt{x^{2} - x + a}} f r o m a n o t h e r s i d e

f^{^{'}} (a) = 2 \times \frac{\frac{1}{\sqrt{a}}}{2 \sqrt{a} + 1} - \frac{4}{4 a - 1} = 2 \times \frac{1}{\sqrt{a} (2 \sqrt{a} + 1)} - \frac{4}{4 a - 1}

= \frac{2}{2 a + \sqrt{a}} - \frac{4}{4 a - 1} \Rightarrow

- \frac{1}{2} \int_{0}^{1} \frac{d x}{(x^{2} - x + a) \sqrt{x^{2} - x + a}} = \frac{2}{2 a + \sqrt{a}} - \frac{4}{4 a - 1} a l s o w e g e t

\int_{0}^{1} \frac{d x}{(x^{2} - x + a) \sqrt{x^{2} - x + a}} = \frac{- 4}{2 a + \sqrt{a}} + \frac{8}{4 a - 1}