Evaluate $\int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $

Question

Evaluate the indefinite integral:

$\displaystyle \int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $

(using substitution)

The answer is: $\frac{2}{11} \sqrt{sec(11 x)} + C$

I don't get where $11$ in $\frac{2}{11}$ comes from

My solution:

u = sec($11x$)

du = sec(11x) $\cdot$ tan(11x) dx

Making substitution:

$\displaystyle \int \frac{1}{\sqrt{u}} du$

Evaluating integral:

$\frac {u^\frac{1}{2}}{\frac {1}{2}} -> 2 \cdot \sqrt {sec(11x)} + C$

As you can see, there shouldn't 11 in the answer... but how come there is?

Answer 1

Your $du$ is wrong. It should have an extra 11 in it. :)

Answer 2

$du = sec(11x) tan (11x) (11dx)$

Answer 3

Just to make things easier on my eyes (I hate all of the 11's) i'd start off with:

$let\:v = 11x,\: dv = 11dx$

$$\frac{1}{11}\int{\frac{\sec v \:\tan v}{\sqrt{\sec v}}}\:dv$$

$let\: u=\sec v,\: du = \sec v\:\tan v\:dv$

$$\frac{1}{11}\int{\frac{1}{\sqrt{u}}}\:du$$

You can take it from here.