7x-6-x-2-25-x-3-2-4-dx-

Question Number 35909 by ajfour last updated on 25/May/18

\int \frac{7 x - 6}{(x^{2} + 25) \sqrt{{(x - 3)}^{2} + 4}} d x = ?

Commented by tanmay.chaudhury50@gmail.com last updated on 26/May/18

t h i s n u t i s h a r d t o c r a c k \dots i a m f i g h t i n g . . h o p e

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

Commented by rahul 19 last updated on 26/May/18

Y e s s i r!

I f t h e r e i s n o (7 x - 6) i n n u m e r a t o r

t h e n i t i s c a k e w a l k!

Commented by prof Abdo imad last updated on 26/May/18

t b e r e i s n o c a k e n i c h o c o l a t i n m a t h s \dots

Commented by abdo mathsup 649 cc last updated on 27/May/18

c h a n g e m e n t x - 3 = 2 s h t g i v e

I = \int \frac{7 (3 + 2 s h t) - 6}{{{(2 s h t + 3)}^{2} + 25} 2 c h t} 2 c h t d t

I = \int \frac{14 s h t + 15}{{4 {s h}^{2} t + 12 s h t + 34)} d t

= \int \frac{14 \frac{e^{t} - e^{- t}}{2} + 15}{4 {(\frac{e^{t} - e^{- t}}{2})}^{2} + 12 \frac{e^{t} - e^{- t}}{2} + 34} d t

= \int \frac{7 e^{t} - 7 e^{- t} + 15}{e^{2 t} + e^{- 2 t} - 2 + 6 e^{t} - 6 e^{- t} + 34} d t

= \int \frac{7 e^{t} - 7 e^{- t} + 15}{e^{2 t} + e^{- 2 t} + 6 e^{t} - 6 e^{- t} + 32} d t c h a n g e m e n t

e^{t} = x g i v e

I = \int \frac{7 x - \frac{7}{x} + 15}{x^{2} + \frac{1}{x^{2}} + 6 x - \frac{6}{x} + 32} \frac{d x}{x}

= \int \frac{7 x^{2} + 15 x - 7}{x^{4} + 1 + 6 x^{3} - 6 x + 32 x^{2}} d x

= \int \frac{7 x^{2} + 15 x - 7}{x^{4} + 6 x^{3} + 32 x^{2} - 6 x + 1} d x t h e r o o t s o f

p o l y n o m p (x) = x^{4} + 6 x^{3} + 32 x^{2} - 6 x + 1 a r e

t h e c o m p l e x z_{1} \sim - 3, 093 + 4, 853 i

z_{2} \sim - 3, 093 - 4, 853 i = {\bar{z}}_{1}

z_{3} \sim 0, 093 + 0, 146 i

z_{4} \sim 0, 093 - 0, 146 i = {\bar{z}}_{3}

p (x) \sim (z -_{} z_{1}) (x - {\bar{z}}_{1}) (x - z_{2}) (x - {\bar{z}}_{2})

\sim (x^{2} + 2.3, 093 x + \sqrt{{(3, 093)}^{2} + {(4, 853)}^{2}}) .

(x^{2} + 2.0, 093 x + \sqrt{{(0, 093)}^{2} + {(0, 146)}^{2}})

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

d e c o m p o s i n g t h e f r a c t i o n

F (x) = \frac{7 x^{2} + 15 x - 7}{p (x)} a t f o r m

F (x) = \frac{a x + b}{x^{2} - 2 R e (z_{1}) x + ∣ z_{1} ∣^{2}} + \frac{c x + d}{x^{2} - 2 R e (z_{2}) x + ∣ z_{2} ∣^{(}}

\dots .

Answered by tanmay.chaudhury50@gmail.com last updated on 26/May/18

\int \frac{7 x - 6}{(x^{2} + 25) \sqrt{x^{2} - 6 x + 13}} d x

l e t t^{2} = \frac{x^{2} - 6 x + 13}{x^{2} + 25}

2 l n t = l n (x^{2} - 6 x + 13) - l n (x^{2} + 25)

\frac{2}{t} d t = {\frac{2 x - 6}{(x^{2} - 6 x + 13)} - \frac{2 x}{x^{2} + 25}} d x

\frac{2}{t} d t = \frac{2 x^{3} + 50 x - 6 x^{2} - 150 - 2 x^{3} + 12 x^{2} - 26 x}{(x^{2} + 25) (x^{2} - 6 x + 13)} d x

\frac{2}{t} d t = \frac{6 x^{2} + 24 x - 150}{(x^{2} + 25) (x^{2} - 6 x + 13)} d x

d x = \frac{2}{t} \times \frac{(x^{2} + 25) (x^{2} - 6 x + 13)}{6 (x^{2} + 4 x - 25)} d t

\int \frac{7 x - 6}{(x^{2} + 25) \sqrt{x^{2} - 6 x + 13}} \frac{2}{t} \times \frac{(x^{2} + 25) (x^{2} - 6 x + 13}{6 (x^{2} + 4 x - 25)}

c o n t d p u t i n g v a l u e o f t

\int \frac{7 x - 6}{(x^{2} + 25) \sqrt{x^{2} - 6 x + 13}} \times \frac{2 \sqrt{x^{2} + 25}}{\sqrt{x^{2} - 6 x + 13}} \times

\frac{(x^{2} + 25) (x^{2} - 6 x + 13)}{6 (x^{2} + 4 x - 25)} d x

\frac{1}{3} \int \frac{(7 x - 6) \sqrt{x^{2} + 25}}{x^{2} + 4 x - 25} d x

c o n t d

Commented by ajfour last updated on 26/May/18

t h a n k s f o r t h e a t t e m p t s i r .

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