AP EAMCET · Maths · Indefinite Integration
If \(\int\left(\frac{4 e^x+6 e^{-x}}{9 e^x-4 e^{-x}}\right) d x=A x+B \log \left|\left(9 e^{2 x}-4\right)\right|+C\), then \((\mathrm{A}, \mathrm{B})=\)
- A \(\left(\frac{3}{2}, \frac{35}{36}\right)\)
- B \(\left(\frac{-3}{2}, \frac{-35}{36}\right)\)
- C \(\left(\frac{-3}{2}, \frac{35}{36}\right)\)
- D \(\left(\frac{3}{2}, \frac{-35}{36}\right)\)
Answer & Solution
Correct Answer
(B) \(\left(\frac{-3}{2}, \frac{-35}{36}\right)\)
Step-by-step Solution
Detailed explanation
Let \(I=\int \frac{4 e^x+6 e^{-x}}{9 e^x-4 e^{-x}} d x=\int \frac{4 e^{2 x}+6}{9 e^{2 x}-4} d x\) Let \(4 e^{2 x}+6=A\left(9 e^{2 x}-4\right)+B\left(18 e^{2 x}\right)\) \(4 e^{2 x}+6=(9 A+18 B) e^{2 x}-4 A\) Comparing both sides we get, \(A=-\frac{3}{2}, B=\frac{35}{36}\)…
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