If int left( frac{4 e^x + 6 e^{-x}}{9 e^x - 4 | vedclass

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Question

(C) Let $I = \int \frac{4 e^x + 6 e^{-x}}{9 e^x - 4 e^{-x}} d x = \int \frac{4 e^{2 x} + 6}{9 e^{2 x} - 4} d x$. We express the numerator as $4 e^{2 x

Vedclass · Accepted Answer

$(-\frac{3}{2}, \frac{35}{36})$

Vedclass · Answer

(C) Let $I = \int \frac{4 e^x + 6 e^{-x}}{9 e^x - 4 e^{-x}} d x = \int \frac{4 e^{2 x} + 6}{9 e^{2 x} - 4} d x$. We express the numerator as $4 e^{2 x} + 6 = A(9 e^{2 x} - 4) + B \frac{d}{d x}(9 e^{2 x} - 4)$. $4 e^{2 x} + 6 = A(9 e^{2 x} - 4) + B(18 e^{2 x})$. Equating the coefficients of $e^{2 x}$ and the constant terms: $9 A + 18 B = 4$ and $-4 A = 6$. From $-4 A = 6$,we get $A = -\frac{3}{2}$. Substituting $A$ into the first equation: $9(-\frac{3}{2}) + 18 B = 4 \Rightarrow -\frac{27}{2} + 18 B = 4 \Rightarrow 18 B = 4 + \frac{27}{2} = \frac{35}{2} \Rightarrow B = \frac{35}{36}$. Thus,$I = \int \left( A + B \frac{18 e^{2 x}}{9 e^{2 x} - 4} \right) d x = A x + B \log |9 e^{2 x} - 4| + C$. Therefore,$(A, B) = (-\frac{3}{2}, \frac{35}{36})$.

If $\int \left( \frac{4 e^x + 6 e^{-x}}{9 e^x - 4 e^{-x}} \right) d x = A x + B \log |9 e^{2 x} - 4| + C$,then $(A, B) = $

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