First slide

Methods of integration

Question

Let $f (x)$ be a function satisfying $f^{'} (x) = f (x)$ and
$f (0) = 2 .$ Then $\int \frac{f (x)}{3 + 4 f (x)} d x$

Moderate

A

$(1 / 4) \log (3 + 8 e^{x}) + C$
B

$(1 / 2) \log (3 + 5 e^{x}) + C$
C

$(1 / 4) \log (3 + 4 e^{2 x}) + C$
D

none of these

Solution

First show that $f (x) = 2 e^{x}$ . Now,
$\begin{array}{l} I = \int \frac{2 e^{x}}{3 + 8 e^{x}} d x \\ = \frac{1}{4} \log (3 + 8 e^{x}) + C \end{array}$

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