Question-68767

Question Number 68767 by aliesam last updated on 15/Sep/19

Commented by ~ À ® @ 237 ~ last updated on 15/Sep/19

L e t n a m e d i t f_{n} (x)

L e t n a m e d \forall a > 0 g (a, x) = \int \frac{d x}{a + x^{2}}

S o g (a, x) = \frac{1}{a} \int \frac{d x}{1 + \frac{x^{2}}{a}} = \frac{1}{\sqrt{a}} \int \frac{d (\frac{x}{\sqrt{a}})}{1 + {(\frac{x}{\sqrt{a}})}^{2}}

g (a, x) = \frac{a r c t a n (\frac{x}{\sqrt{a}})}{\sqrt{a}} + c

N o w j u s t a s c e r t a i n t h a t

\frac{\partial g (a, x)}{\partial a} = - \int \frac{d x}{{(a + x^{2})}^{2}}

\frac{\partial^{2} g (a, x)}{\partial a^{2}} = \int \frac{2 d x}{{(a + x^{2})}^{3}}

\dots \dots \frac{\partial^{n} g (a, x)}{\partial a^{n}} = \int \frac{{(- 1)}^{n} (n - 1)}{{(a + x^{2})}^{n}} d x

F i n a l l y

\int \frac{d x}{{(a + x^{2})}^{n}} = \frac{{(- 1)}^{n}}{n - 1} \frac{\partial^{n - 1} g (a, x)}{\partial a^{n - 1}}

f_{n} (x) = \frac{{(- 1)}^{n}}{n - 1} \frac{\partial^{n - 1} g}{\partial a^{n - 1}} (1, x)