If-4e-x-6e-x-9e-x-4e-x-dx-Ax-B-log-9e-2x-4-C-then-A-B-C-

Question Number 53311 by gunawan last updated on 20/Jan/19

If \int \frac{4 e^{x} + 6 e^{- x}}{9 e^{x} - 4 e^{- x}} d x = A x + B \log (9 e^{2 x} - 4) + C

then

A = \dots

B = \dots

C = \dots

Commented by maxmathsup by imad last updated on 20/Jan/19

l e t I = \int \frac{4 e^{x} + 6 e^{- x}}{9 e^{x} - 4 e^{- x}} d x \Rightarrow I =_{e^{x} = t} \int \frac{4 t + 6 t^{- 1}}{9 t - 4 t^{- 1}} \frac{d t}{t} = \int \frac{4 t + 6 t^{- 1}}{9 t^{2} - 4} d t

= \int \frac{4 t^{2} + 6}{9 t^{3} - 4 t} d t l e t d e c o m p o s e F (t) = \frac{4 t^{2} + 6}{t (9 t^{2} - 4)} = \frac{4 t^{2} + 6}{t (3 t - 2) (3 t + 2)}

= \frac{a}{t} + \frac{b}{3 t - 2} + \frac{c}{3 t + 2}

a = {l i m}_{t \to 0} t F (t) = - \frac{3}{2}

b = {l i m}_{t \to \frac{2}{3}} (3 t - 2) F (t) = \frac{4 . \frac{4}{9} + 6}{\frac{2}{3} (4)} = \frac{\frac{8}{9} + 3}{\frac{4}{3}} = \frac{35}{9} . \frac{3}{4} = \frac{35}{12}

c = {l i m}_{t \to - \frac{2}{3}} (3 t + 2) F (t) = \frac{4 . \frac{4}{9} + 6}{(- \frac{2}{3}) (- 4)} = \frac{35}{12} \Rightarrow

I = - \frac{3}{2} \int \frac{d t}{t} + \frac{35}{12} \int \frac{d t}{3 t - 2} + \frac{35}{12} \int \frac{d t}{3 t + 2} + c

= - \frac{3}{2} l n ∣ t ∣ + \frac{35}{36} l n ∣ 3 t - 2 ∣ + \frac{35}{36} l n ∣ 3 t + 2 ∣ + c

= - \frac{3}{2} l n ∣ t ∣ + \frac{35}{36} l n ∣ 9 t^{2} - 4 ∣ + c

= - \frac{3}{2} x + \frac{35}{36} l n (9 e^{2 x} - 4) + c \Rightarrow a = - \frac{3}{2} a n d b = \frac{35}{36}

c \to c o n s t a n t o f i n t e g r a t i o n .

Answered by tanmay.chaudhury50@gmail.com last updated on 20/Jan/19

4 e^{x} + 6 e^{- x} = P (9 e^{x} - 4 e^{- x}) + Q \times \frac{d}{d x} (9 e^{x} - 4 e^{- x})

= P (9 e^{x} - 4 e^{- x}) + Q (9 e^{x} + 4 e^{- x})

= e^{x} (9 P + 9 Q) + e^{- x} (- 4 P + 4 Q)

9 P + 9 Q = 4 \times 4

- 4 P + 4 Q = 6 \times 9

36 P + 36 Q = 16

- 36 P + 36 Q = 54

72 Q = 70 [Q = \frac{35}{36}] 4 P = 4 \times \frac{35}{36} - 6

4 P = \frac{140 - 216}{36} [P = \frac{- 76}{4 \times 36} = \frac{- 19}{36}]

\int \frac{4 e^{x} + 6 e^{- x}}{9 e^{x} - 4 e^{- x}} d x

= \int \frac{P (9 e^{x} - 4 e^{- x}) + Q \times \frac{d}{d x} (9 e^{x} - 4 e^{- x})}{9 e^{x} - 4 e^{- x}} d x

= P \int d x + Q \int \frac{d (9 e^{x} - 4 e^{- x})}{9 e^{x} - 4 e^{- x}}

= P x + Q l n (9 e^{x} - 4 e^{- x}) + c_{1}

= P x + Q [l n (\frac{9 e^{2 x} - 4}{e^{x}})] + c_{1}

= P x + Q l n (9 e^{2 x} - 4) - Q l n (e^{x}) + c_{1}

= P x + Q l n (9 e^{2 x} - 4) - Q x + c_{1}

= (P - Q) x + Q l n (9 e^{2 x} - 4) + c_{1}

= (\frac{- 19}{36} - \frac{35}{36}) x + \frac{35}{36} l n (9 e^{2 x} - 4) + c_{1}

= (\frac{- 54}{36}) x + \frac{35}{36} l n (9 e^{2 x} - 4) + c_{1}

A \to \frac{- 54}{36} B \to \frac{35}{36} C \to c_{1}

p l s c h e c k s t e p s m i s t a k e i f a n y