1. ## antiderivative

Hi I need some help with this antiderivative.

cos(5x)+sinh(2x)

2. ## Antiderivative

First, the antiderivative of a sum is the sum of the antiderivatives:
$\int \cos (5x) + \sinh (2x) dx = \int \cos (5x) dx + \int \sinh (2x) dx$

Then we look at $\cos (5x)$. The antiderivative of $\cos x$ is $\sin x$. By the chain rule

$\frac{d}{dx} \sin (5x) = \cos (5x) \cdot \frac{d}{dx} (5x) = 5 \cos (5x)$,
thus $\frac{d}{dx} (\tfrac{1}{5}\sin (5x)) = \tfrac{1}{5} \cdot 5 \cos (5x) = \cos (5x)$, and the antiderivative of $\cos(5x)$ is $\tfrac{1}{5}\sin (5x)$.

Similar reasoning from the antiderivative of $\sinh x$ will give you the other half of the antiderivative.

--Kevin C.