Originally Posted by

**craig** Hi, could someone please help me out on this question.

Using the substitution $\displaystyle x = \frac{3}{sinh \theta}$, find the exact value of:

$\displaystyle \int_{4}^{3\sqrt{3}} \frac{1}{x\sqrt{x^2 + 9}} dx$

If $\displaystyle x = \frac{3}{sinh \theta}$, then $\displaystyle \frac{dx}{d\theta} = \frac{-3cosh \theta}{sinh^2 \theta}$.

Putting in the substitution, I get $\displaystyle \int_{4}^{3\sqrt{3}} \frac{1}{\frac{3}{sinh \theta}\sqrt{\frac{9}{sinh^2 \theta} + 9}} \times \frac{-3cosh \theta}{sinh^2 \theta} d\theta$.

From this I moved the $\displaystyle sinh \theta$ to the top of the first fraction and cancelled with one of the $\displaystyle sinh^2 \theta$ on the denominator of the second fraction to give me this.

$\displaystyle \int_{4}^{3\sqrt{3}} \frac{1}{3\sqrt{\frac{9}{sinh^2 \theta} + 9}} \times \frac{-3cosh \theta}{sinh \theta} d\theta$

I have *no* idea what to do next, would really appreciate some help?

Thanks in advance