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Question Number 55274 by maxmathsup by imad last updated on 20/Feb/19

l e t φ (a) = \int_{1}^{\sqrt{3}} a r c t a n (\frac{a}{x}) d x

1) c a l c u l a t e φ (a) i n t e r m s o f a

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

3) d e t e r m i n e φ^{(n)} (a) a t f o r m o f i n t e g r a l .

4) f i n d t h e v a l u e o f \int_{1}^{\sqrt{3}} a r c t a n (\frac{2}{x}) d x .

Commented by maxmathsup by imad last updated on 24/Feb/19

1) c h a n g e m e n t \frac{a}{x} = t g i v e x = \frac{a}{t} \Rightarrow φ (a) = \int_{a}^{\frac{a}{\sqrt{3}}} a r c t a n (t) (\frac{- a}{t^{2}}) d t \Rightarrow

\frac{φ (a)}{a} = \int_{\frac{a}{\sqrt{3}}}^{a} \frac{a r c t a n (t)}{t^{2}} d t b y p a r t s w e g e t \frac{φ (a)}{a} = {[- \frac{1}{t} a r c t a n (t)]}_{\frac{a}{\sqrt{3}}}^{a} + \int_{\frac{a}{\sqrt{3}}}^{a} \frac{1}{t (1 + t^{2})} d t

= \frac{\sqrt{3}}{a} a r c t a n (\frac{a}{\sqrt{3}}) - \frac{a r c t a n (a)}{a} + \int_{\frac{a}{\sqrt{3}}}^{a} \frac{d t}{t (1 + t^{2})} d t b u t

\int_{\frac{a}{\sqrt{3}}}^{a} \frac{d t}{t (1 + t^{2})} d t = \int_{\frac{a}{\sqrt{3}}}^{a} (\frac{1}{t} - \frac{t}{1 + t^{2}}) d t = {[l n (\frac{t}{\sqrt{1 + t^{2}}})]}_{\frac{a}{\sqrt{3}}}^{a} = l n (\frac{a}{\sqrt{1 + a^{2}}})

- l n (\frac{a}{\sqrt{3} (\sqrt{1 + \frac{a^{2}}{3}}}) = l n (\frac{a}{\sqrt{1 + a^{2}}}) - l n (\frac{a}{\sqrt{a^{2} + 3}})

= l n (a) - \frac{1}{2} l n (1 + a^{2}) - l n (a) + \frac{1}{2} l n (a^{2} + 3) = \frac{1}{2} {l n (a^{2} + 3) - l n (a^{2} + 1)} \Rightarrow

\frac{φ (a)}{a} = \frac{\sqrt{3}}{a} a r c t a n (\frac{a}{\sqrt{3}}) - \frac{a r c t a n (a)}{a} + l n (\sqrt{\frac{a^{2} + 3}{a^{2} + 1}}) \Rightarrow

φ (a) = \sqrt{3} a r c t a n (\frac{a}{\sqrt{3}}) - a r c t a n (a) + a l n (\sqrt{\frac{a^{2} + 3}{a^{2} + 1}}) .

Commented by maxmathsup by imad last updated on 24/Feb/19

2) w e h a v e φ^{^{'}} (a) = \int_{1}^{\sqrt{3}} \frac{1}{x (1 + \frac{a^{2}}{x^{2}})} d x = \int_{1}^{\sqrt{3}} \frac{x}{x^{2} + a^{2}} d x .

Commented by maxmathsup by imad last updated on 24/Feb/19

4) w e h a v e φ (a) = \int_{1}^{\sqrt{3}} a r c t a n (\frac{a}{x}) d x \Rightarrow

\int_{1}^{\sqrt{3}} a r c t a n (\frac{2}{x}) d x = φ (2) = \sqrt{3} a r c t a n (\frac{2}{\sqrt{3}}) - a r c t a n (2) + 2 l n (\sqrt{\frac{7}{5}})

= \sqrt{3} a r c t a n (\frac{2}{\sqrt{3}}) - a r c t a n (2) + l n (7) - l n (5) .

Commented by maxmathsup by imad last updated on 24/Feb/19

3) w e h a v e φ^{(1)} (a) = \int_{1}^{\sqrt{3}} \frac{x}{x^{2} + a^{2}} d x \Rightarrow φ^{(n)} (a) = \int_{1}^{\sqrt{3}} \frac{d^{n - 1}}{{d a}^{n - 1}} (\frac{x}{x^{2} + a^{2}}) d x l e t

= \int_{1}^{\sqrt{3}} x {\frac{1}{x^{2} + a^{2}}}_{/ a}^{(n - 1)} d x = \int_{1}^{\sqrt{3}} \frac{1}{2 i} {\frac{1}{a - i x} - \frac{1}{a + i x}}_{/ a}^{(n - 1)} d x

= \frac{1}{2 i} \int_{1}^{\sqrt{3}} {\frac{{(- 1)}^{n - 1} (n - 1)!}{{(a - i x)}^{n}} - \frac{{(- 1)}^{n - 1} (n - 1)!}{{(a + i x)}^{n}}} d x

= \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} \int_{1}^{\sqrt{3}} \frac{{(a + i x)}^{n} - {(a - i x)}^{n}}{{(a^{2} + x^{2})}^{n}} d x

= {(- 1)}^{n - 1} (n - 1)! \int_{1}^{\sqrt{3}} \frac{I m ({(a + i x)}^{n})}{{(a^{2} + x^{2})}^{n}} d x .