dx-x-x-1-x-2-x-3-x-n-

Question Number 47740 by tanmay.chaudhury50@gmail.com last updated on 14/Nov/18

\int \frac{d x}{x (x + 1) (x + 2) (x + 3) \dots (x + n)}

Answered by ajfour last updated on 14/Nov/18

\frac{1}{x (x + 1) (x + 2) \dots (x + r) \dots (x + n)}

= Σ \frac{A_{r}}{x + r}

A_{r} = \frac{1}{{(- 1)}^{r} r! (n - r)!}

\Rightarrow I = \underset{r = 0}{\sum^{n}} \frac{1}{{(- 1)}^{r} r! (n - r)!} \ln (x + r) + c .

Commented by tanmay.chaudhury50@gmail.com last updated on 14/Nov/18

e x c e l l e n t \dots

Commented by ajfour last updated on 14/Nov/18

a s e x a m p l e

I = \int \frac{d x}{x (x + 1) (x + 2)}

= \frac{1}{2} \ln ∣ x ∣ - \ln ∣ x + 1 ∣ + \frac{1}{2} \ln ∣ x + 2 ∣ + c .

Answered by tanmay.chaudhury50@gmail.com last updated on 14/Nov/18

\frac{1}{x (x + 1) (x + 2) (x + 3) \dots (x + n)} = \frac{a_{0}}{x} + \frac{a_{1}}{x + 1} + \frac{a_{2}}{x + 2} + \dots + \frac{a_{n}}{x + n}

s o

1 = a_{0} (x + 1) (x + 2) (x + 3) \dots (x + n) + a_{1} (x) (x + 2) (x + 3) . . (x + n) + \dots

t o f i n d a_{0} p u t x = 0

a_{0} \times n! = 1 a_{0} = \frac{1}{n!}

s i m i l a r l y t o f i n d a_{1} p u t x + 1 = 0

a_{1} (- 1) (1) (2) \dots (n - 1) = 1

a_{1} \times (- 1) \times (n - 1)! = 1

a_{1} = \frac{1}{(- 1) (n - 1)!}

i n t h i s w a y w e f i n d a_{0}, a_{1}, a_{2} \dots a_{n}

s o t h e i n t r e g a t i o n v a l u e i s

a_{0} l n x + a_{1} l n (x + 1) + a_{2} (x + 2) \dots . + a_{n} (x + n) + c

n x t s t e p p u t v a l u e o f a_{0}, a_{1}, a_{2} \dots . a_{n}

i h a v e s o l v e d i n d e t a i l f o r u n d e r s t a n d i n g \dots