Year 12 Calculus: Integration

Integration reverses differentiation and is used to find antiderivatives, accumulated change and areas under curves. The most important first step is choosing the right antiderivative rule.

For indefinite integration, remember the constant of integration. For definite integration, evaluate the antiderivative at the upper and lower limits and subtract.

Worked example

Problem: Evaluate

\int_{1}^{3} (2 x + 1) d x

Find an antiderivative: $\int (2 x + 1) d x = x^{2} + x$ .
Apply the limits: ${[x^{2} + x]}_{1}^{3}$ .
Substitute the upper limit: $3^{2} + 3 = 12$ .
Substitute the lower limit: $1^{2} + 1 = 2$ .
Subtract: $12 - 2 = 10$ .

Answer:

\int_{1}^{3} (2 x + 1) d x = 10

Indefinite versus definite integration

An indefinite integral gives a family of antiderivatives, so it always needs a constant of integration. A definite integral gives a single number because the interval is fixed. Even when the antiderivative step is the same, the final interpretation is different: one is a function family and the other is accumulated change or signed area.

Worked example 2

Problem: Find

\int (3 x^{2} - 4 x + 5) d x

Integrate term by term.
$\int 3 x^{2} d x = x^{3}$ .
$\int - 4 x d x = - 2 x^{2}$ .
$\int 5 d x = 5 x$ .
Add the constant of integration: $x^{3} - 2 x^{2} + 5 x + C$ .

Answer:

x^{3} - 2 x^{2} + 5 x + C

Common mistakes

Forgetting the constant of integration in an indefinite integral.
Using the power rule incorrectly, especially with negative or fractional powers.
Evaluating the upper limit and forgetting to subtract the lower-limit value.

Integration questions become much calmer when you separate the antiderivative step from the interpretation step. First ask, "what function differentiates to this expression?" Then ask what the question wants you to do with that antiderivative: leave it as an indefinite integral, apply bounds, or interpret it as accumulated change. Mixing those stages is one of the main reasons students lose track.

It also helps to keep the power rule connected to differentiation. If differentiating $x^{n + 1}$ produces $(n + 1) x^{n}$ , then integrating $x^{n}$ should reverse that change. Thinking of integration as reversal, rather than memorising it as a separate rule list, makes the technique much more secure.

Revision focus

In calculus, one of the best habits is to decide what role the expression is playing before you touch a rule. Is it a product, a composite function, an implicitly defined curve, or an antiderivative problem? That classification step is not wasted time. It is the step that usually determines whether the next line of working will be short and accurate or long and unstable. Strong calculus work looks calm because the structure is identified early, not because the algebra is magically easier.