1-a-0-calculate-0-a-n-2-x-2-n-2-x-2-2-dx-with-n-integr-2-find-0-n-2-x-2-n-2-x-2-2-dx-3-calculate-n-1-0-n-2-x-2-n-2-x-2-2-dx-

Question Number 32734 by caravan msup abdo. last updated on 01/Apr/18

1) a ⩾ 0 c a l c u l a t e \int_{0}^{a} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} d x w i t h

n i n t e g r

2) f i n d \int_{0}^{\infty} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} d x

3) c a l c u l a t e \sum_{n = 1}^{\infty} \int_{0}^{\infty} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} d x .

Commented by abdo imad last updated on 08/Apr/18

l e t p u t I_{n} (a) = \int_{0}^{a} \frac{n^{2} - x^{2}}{(n^{2} + x^{2})} d x

I_{n} (a) = n^{2} \int_{0}^{a} \frac{d x}{{(n^{2} + x^{2})}^{2}} - \int_{0}^{a} \frac{n^{2} + x^{2} - n^{2}}{(n^{2} + x^{2})} d x

= 2 n^{2} \int_{0}^{a} \frac{d x}{{(n^{2} + x^{2})}^{2}} - \int_{0}^{a} \frac{d x}{n^{2} + x^{2}} . c h . x = n t g i v e

\int_{0}^{a} \frac{d x}{n^{2} + x^{2}} = \int_{0}^{\frac{a}{n}} \frac{n d t}{n^{2} (1 + t^{2})} = \frac{1}{n} a r c t a n (\frac{a}{n}) a l s o

\int_{0}^{a} \frac{d x}{{(n^{2} + x^{2})}^{2}} = \int_{0}^{\frac{a}{n}} \frac{n d t}{n^{4} {(1 + t^{2})}^{2}} = \frac{1}{n^{3}} \int_{0}^{\frac{a}{n}} \frac{d t}{{(1 + t^{2})}^{2}}

=_{t = t a n θ} \frac{1}{n^{3}} \int_{0}^{a r c t a n (\frac{a}{n})} \frac{(1 + {t a n}^{2} θ) d θ}{{(1 + {t a n}^{2} θ)}^{2}}

= \frac{1}{n^{3}} \int_{0}^{a r c t a n (\frac{a}{n})} \frac{d θ}{1 + {t a n}^{2} θ} = \frac{1}{n^{3}} \int_{0}^{a r c t a n (\frac{a}{n})} ({c o s}^{2} θ) d θ

= \frac{1}{2 n^{3}} \int_{0}^{a r c t a n (\frac{a}{n})} (1 + c o s (2 θ)) d θ

= \frac{a r c t a n (\frac{a}{n})}{2 n^{3}} + \frac{1}{4 n^{3}} {[s i n (2 θ)]}_{0}^{a r c t a n (\frac{a}{n})}

= \frac{a r c t a n (\frac{a}{n})}{2 n^{3}} + \frac{1}{4 n^{3}} s i n (2 a r c t a n (\frac{a}{n})) \Rightarrow

I_{n} (a) = \frac{a r c t a n (\frac{a}{n})}{n} + \frac{1}{2 n} s i n (2 a r c t a n (\frac{a}{n})) - \frac{1}{n} a r c t a n (\frac{a}{n})

I_{n} (a) = \frac{1}{2 n} s i n (2 a r c t a n (\frac{a}{n})) .

Commented by abdo imad last updated on 08/Apr/18

w e h a v e s i n (2 a r c t a n (\frac{a}{n})) = 2 s i n (a r c t a n (\frac{a}{n})) c o s (a r c t s n (\frac{a}{n}))

b u t w e h a v e t h e f r m u l a e s i n (a r c t a n x) = \frac{x}{\sqrt{1 + x^{2}}}

c o s (a r c t a n x) = \frac{1}{\sqrt{1 + x^{2}}} \Rightarrow

I_{n} (a) = \frac{1}{n} \frac{\frac{a}{n}}{\sqrt{1 + \frac{a^{2}}{n^{2}}}} \frac{1}{\sqrt{1 + \frac{a^{2}}{n^{2}}}} = \frac{a}{(1 + \frac{a^{2}}{n^{2}})} = \frac{{a n}^{2}}{n^{2} + a^{2}}

Commented by abdo imad last updated on 08/Apr/18

2) \int_{0}^{\infty} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} = {l i m}_{a \to + \infty} I_{n} (a) = {l i m}_{a \to + \infty} \frac{{a n}^{2}}{n^{2} + a^{2}}

= {l i m}_{a \to + \infty} \frac{n^{2}}{a} = 0 .

Answered by hknkrc46 last updated on 04/Apr/18

1) x = ntanu \Rightarrow dx = n (1 + \tan^{2} u) du = {nsec}^{2} udu

x \to a, u \to \arctan (\frac{a}{n}) \land x \to 0, u \to 0

\int_{0}^{a} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} dx = \int_{0}^{\arctan (\frac{a}{n})} \frac{n^{2} - {(ntanu)}^{2}}{{(n^{2} + {(ntanu)}^{2})}^{2}} {nsec}^{2} udu

= \int_{0}^{\arctan (\frac{a}{n})} \frac{n^{2} - n^{2} \tan^{2} u}{{(n^{2} + n^{2} \tan^{2} u)}^{2}} {nsec}^{2} udu

= \int_{0}^{\arctan (\frac{a}{n})} \frac{n^{2} (1 - \tan^{2} u)}{{(n^{2} (1 + \tan^{2} u))}^{2}} {nsec}^{2} udu

= \int_{0}^{\arctan (\frac{a}{n})} \frac{\frac{n^{2} \cos 2 u}{\cos^{2} u}}{n^{4} \sec^{4} u} {nsec}^{2} udu

= \int_{0}^{\arctan (\frac{a}{n})} \frac{\cos 2 u}{n} du = \frac{\sin 2 u}{2 n} ∣_{0}^{\arctan (\frac{a}{n})} = \frac{\sin (2 \arctan (\frac{a}{n}))}{2 n}

= \frac{\sin (\arctan (\frac{a}{n})) \cos (\arctan (\frac{a}{n}))}{n} = \frac{a}{a^{2} + n^{2}}

★ \arctan (\frac{a}{n}) = ψ \Rightarrow \tan ψ = \frac{a}{n} \Rightarrow \sin ψ = \frac{a}{\sqrt{a^{2} + n^{2}}} \land \cos ψ = \frac{n}{\sqrt{a^{2} + n^{2}}}

2) \int_{0}^{\infty} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} dx = \lim_{a \to \infty} \int_{0}^{a} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} dx = \lim_{a \to \infty} \frac{a}{a^{2} + n^{2}} = 0

3) \sum_{n = 1}^{\infty} \int_{0}^{\infty} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} dx = \lim_{k \to \infty} \sum_{n = 1}^{k} \int_{0}^{\infty} \frac{n^{2} - x^{2}}{{(n^{2} + x^{2})}^{2}} dx = 0

Facebook Tweet Pin