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I have a calculus II final coming up and this question came up in a past final exam:

$$\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x – 16}} = \frac{\pi}{k},$$

where $k$ is a constant. Find $k$.

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- Why does $\int_1^\infty \sin(x \log x)\,\mathrm{d}x$ converge?
- Taylor expansion around infinity of a fraction

My progress so far:

$$\int_{6}^{16} \frac{dx}{\sqrt{(x – 1)}(x + 4)} = \frac{\pi}{k}$$

The answer is: $k = 6\sqrt{5}$

I do not know where to from from this step. Any helps or hints will be greatly appreciated! Thank you.

EDIT: $(x + 4)$ not $(x + 2)$ in the denominator.

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- Why it is absolutely mistaken to cancel out differentials?

Use substitution, put say $u = \sqrt{x-1}$ and then see what happens. It is useful to know that the derivative of $\tan^{-1} x $ is $\frac{1}{x^2+1}$.

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