Question

Find the integral. \(\int \frac{\cosh x}{\sqrt{9-\sinh ^{2} x}} d x\)

Short answer

The answer is \(\sinh^{-1}\frac{x}{3} + C\), where C is the constant of integration.

Step-by-step solution

Identifying a pattern

The integral in question is similar to the derivative of the inverse hyperbolic sine (\(\sinh^{-1}\)) function. The derivative of \(\sinh^{-1}x\) is \(\frac{1}{\sqrt{1+x^{2}}}\), which is similar to the integrand provided. Rewriting \(\frac{1}{\sqrt{9}}\) as \(\sqrt{\frac{1}{9}}\) for simplicity, the integral now has the form \(\int \sqrt{\frac{1}{9}} \cosh x d x\). The integral may, thus, be re-written as \(\int \cosh x d (\sinh^{-1} \frac{x}{3})\).

Applying the substitution method

Use the substitution \(u = \sinh^{-1}\frac{x}{3}\) which allows us to write \(\cosh x dx = du\). So, the integral now becomes \(\int du\).

Evaluating the integral

The integral of \(du\) is simply \(u\). Since we had previously substituted, leftover \(u\) values should now be transformed back to original form, i.e., \(u = \sinh^{-1}\frac{x}{3}\).