Integration of Hyperbolic Functions

Example 1:

$$\int \sinh 3x \, dx $$

Solution:

$$\int \sinh 3x \, dx $$ $$\textup{Let}\,\, u=3x$$ $$du=3dx$$ $$\frac{du}{3}=dx$$ $$\int \sinh 3x \, dx = \int \sinh u \, \frac{dx}{3} $$ $$= \frac{1}{3} \int \sinh u\, du$$ $$\because\int \sinh x \, dx = \int \cosh x \,dx$$ $$\therefore\int \sinh 3x \, dx = \frac{1}{3} \cosh u + C$$ $$= \frac{1}{3} \cosh 3x + C$$