Find antiderivatives of exponential and trigonometric functions and apply them in definite integrals.

What this dot point is asking

Integration is antidifferentiation, so every derivative rule you learned in Unit 3 reverses into an integration rule. This dot point collects the antiderivatives of the exponential and trigonometric functions, completing the toolkit alongside the power and reciprocal rules.

Reversing each derivative

Because $\dfrac{d}{dx}(e^{kx}) = ke^{kx}$, integrating $e^{kx}$ must divide by $k$ to cancel that factor. The trigonometric antiderivatives follow the same logic, with the added care that integrating sine introduces a minus sign.

Checking by differentiating back

The reliable way to confirm any antiderivative is to differentiate your answer and check you recover the original integrand. For instance, differentiating $-\tfrac{1}{3}\cos(3x)$ gives $-\tfrac{1}{3}\cdot(-3\sin(3x)) = \sin(3x)$, which confirms the antiderivative of $\sin(3x)$.

Worked example

Where these antiderivatives are used

These rules let you find areas under exponential and trigonometric curves, recover totals from exponential or oscillating rates of change, and compute the mean of a continuous random variable whose density involves $e^{-kx}$. They also combine with the area-between-curves method whenever one of the boundaries is an exponential or trigonometric graph.

Summary

Reverse each derivative: dividing by $k$ for $e^{kx}$ and for $\sin(kx)$ or $\cos(kx)$, and remembering that the integral of sine carries a minus sign while the integral of cosine does not. Always verify by differentiating back. With these antiderivatives added to the power and reciprocal rules, you can integrate any standard function in the course and apply it in definite integrals.