Question

The anti derivative of $f(x) = \frac{1}{3 + 5\sin x + 3\cos x}$ whose graph passes through the point $(0,0)$ is

• A. $\frac{1}{5}\left( \log\left| ⥂ 1 - \frac{5}{3}\tan x/2 \right| \right)$
• B. $\frac{1}{5}\left( \log\left| ⥂ 1 + \frac{5}{3}\tan x/2 \right| \right)$
• C. $\frac{1}{5}\left( \log\left| ⥂ 1 + \frac{5}{3}\cot x/2 \right| \right)$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{5}\left( \log\left| ⥂ 1 + \frac{5}{3}\tan x/2 \right| \right)$

Answer 2

$y = \int_{}^{}{\frac{dx}{(3 + 5\sin x + 3\cos x)} = \int_{}^{}\frac{\sec^{2}x/2dx}{10\tan x/2 + 6}}$ $= \frac{1}{5}\log(5\tan x/2 + 3) + c$ $= \frac{1}{5}\log\left| ⥂ \frac{5}{3}\tan x/2 + 1 \right| + c$

Passes through$(0,0)$

$\therefore c = 0$ then $y = \frac{1}{5}\log\left| 1 + \frac{5}{3}\tan x/2 \right|$.