Integration of Hyperbolic & Inverse Hyperbolic Functions

Find the following integrals:

a. $$\int \frac{\sinh\left(\frac{1}{\sqrt{x}}\right)}{(\sqrt{x})^3} , dx$$

b. $$\int - \cosh x \cdot e^x , dx$$

c. $$\int x \sqrt{(\ln 2x)^2 + 9} , dx$$

d. $$\int \frac{\tan^{-1}\left(\frac{x}{2}\right)}{x^2 + 4} , dx$$

e. $$\int \frac{e^x}{\sqrt{e^{2x} + 1}} , dx$$

Find:

a. $$\int_{4}^{6} \frac{1}{\sqrt{x^2 - 9}} , dx$$

b. $$\int_{3}^{6} \frac{1}{\sqrt{x^2 + 9}} , dx$$

c. $$\int_{0}^{\frac{1}{2}} \frac{\sin^{-1}(2x)}{\sqrt{1 - 4x^2}} , dx$$

d. $$\int_{0}^{\frac{1}{3}} \frac{\sinh^{-1}(3x)}{\sqrt{9x^2 + 1}} , dx$$

e. $$\int_{\frac{1}{2}}^{1} \frac{\cosh^{-1}(2x)}{\sqrt{4x^2 - 1}} , dx$$

f. $$\int \ln(\sinh x)^2 , dx$$

Source: FAC1004 Tutorial 10 25-26.pdf