Question

The first order derivative of $ f(x) $ is given by $ f'(x) = sec^4x + 5 $ with $ f(0) = 0 $ . Then $ f(x) = $

• A. $ tan\,x+\frac{1}{3}tan^{3}\,x+5x+2 $
• B. $ tan\,x+\frac{1}{3}tan^{3}\,x+5x $
• C. $ tan\,x+\frac{1}{3}tan^{3}\,x $
• D. $ \frac{1}{3} tan^{3}\,x+5x $

Answer 1

Answer: (B)

We have, $f'(x) = sec^4 x + 5$
$\Rightarrow \int f'(x) dx = \int sec^4 x\,dx + \int 5\, dx$
$\Rightarrow f(x) = tan\,x + \frac{1}{3} tan^3 \,x + 5x +c$
Now, $f(0) = tan(0) + \frac{1}{3} tan^3(0) + 5 \times 0 + c = 0$
$ \Rightarrow c = 0$