Find antiderivatives of standard functions, include the constant of integration, and determine a particular antiderivative from an initial condition

What this dot point is asking

SCSA Unit 4 begins integral calculus by reversing the differentiation of Unit 3. This dot point asks you to find antiderivatives of standard functions, always include the constant of integration, and use a condition to determine a particular solution. It is examined in both the calculator-free and calculator-assumed sections.

Reversing differentiation

If $F'(x)=f(x)$, then $F$ is an antiderivative of $f$. Because the derivative of a constant is zero, adding any constant gives another valid antiderivative, so the general antiderivative includes $+c$.

The $\dfrac{1}{k}$ factors reverse the chain-rule factor that differentiation of $e^{kx}$, $\sin(kx)$ or $\cos(kx)$ would have produced.

The constant of integration

The constant $c$ is not optional: the indefinite integral represents an entire family of parallel curves differing only by a vertical shift. Omitting it both loses a mark and makes it impossible to apply an initial condition.

Finding a particular antiderivative

When extra information is given, such as a point on the curve, substitute it to solve for $c$ and select the one antiderivative that fits.