Find antiderivatives of standard functions including polynomial, exponential and trigonometric forms, evaluate definite integrals using the Fundamental Theorem of Calculus, and recognise the definite integral as the limit of a Riemann sum

What this dot point is asking

QCAA wants you to recognise integration as the reverse of differentiation, find antiderivatives of all standard Methods functions, evaluate definite integrals using the Fundamental Theorem of Calculus (FTC), and connect the definite integral to the limit of a Riemann sum. Integration underlies the rest of Topic 2 (area, average value, kinematics) and appears on every Paper 1 and Paper 2.

The answer

Standard antiderivatives

The constant $C$ is required on every indefinite integral.

The minus sign on $\int \sin x \, dx$ is the Paper 1 trap that mirrors the minus sign on $\frac{d}{dx} \cos x$. These two are paired; remember them together.

Linear inside argument

If the argument is linear ($ax + b$), divide by the coefficient of $x$.

These are not new rules. They are the chain rule run in reverse, with the linear inside argument simple enough that the $\frac{1}{a}$ factor is the only adjustment needed.

The Fundamental Theorem of Calculus

If $F$ is any antiderivative of $f$ (so $F'(x) = f(x)$), then

A second statement: if $G(x) = \displaystyle \int_a^x f(t) \, dt$, then $G'(x) = f(x)$. In short, differentiation and integration are inverse operations.

The FTC turns the geometric problem (area under a curve) into the algebraic problem (evaluate an antiderivative at two points and subtract).

The Riemann sum definition

The definite integral $\int_a^b f(x) \, dx$ is defined as the limit of a Riemann sum:

where the interval $[a, b]$ is split into $n$ subintervals of width $\Delta x = \frac{b - a}{n}$ and $x_i^*$ is a sample point in the $i$-th subinterval. When $f(x) \geq 0$, this limit equals the area under the curve from $x = a$ to $x = b$. When $f(x)$ is negative, the integral counts that area as negative.

For Methods, QCAA expects you to recognise this definition and to use it to interpret what a definite integral represents (an accumulation), without needing to compute Riemann sums by hand at scale.

Properties of the definite integral

These properties speed up Paper 1 evaluation.

• Linearity: $\int_a^b \bigl( c_1 f(x) + c_2 g(x) \bigr) dx = c_1 \int_a^b f + c_2 \int_a^b g$.
• Reversed limits: $\int_b^a f(x) \, dx = -\int_a^b f(x) \, dx$.
• Splitting: $\int_a^c f(x) \, dx = \int_a^b f + \int_b^c f$.
• Zero-width: $\int_a^a f(x) \, dx = 0$.

Worked examples

Standard indefinite integral

$\int (4 x^3 - 6 x + 2) \, dx = x^4 - 3 x^2 + 2 x + C.$

Linear inside argument

$\int (2 x + 1)^4 \, dx = \dfrac{(2 x + 1)^5}{2 \cdot 5} + C = \dfrac{(2 x + 1)^5}{10} + C.$

Exponential with linear argument

$\int e^{-3 x} \, dx = \dfrac{e^{-3 x}}{-3} + C = -\dfrac{1}{3} e^{-3 x} + C.$

Definite integral

Evaluate $\int_1^3 (x^2 + 2 x) \, dx$.

Antiderivative: $F(x) = \dfrac{x^3}{3} + x^2$.

$F(3) - F(1) = \bigl( 9 + 9 \bigr) - \bigl( \tfrac{1}{3} + 1 \bigr) = 18 - \tfrac{4}{3} = \dfrac{54 - 4}{3} = \dfrac{50}{3}.$

Definite integral with the FTC reverse

If $G(x) = \displaystyle \int_2^x e^{t^2} \, dt$, then $G'(x) = e^{x^2}$. No antiderivative needed; the FTC reads off the derivative directly.

Negative area

Evaluate $\int_{-1}^{1} x^3 \, dx$.

Antiderivative: $\frac{x^4}{4}$. Evaluating: $\frac{1}{4} - \frac{1}{4} = 0$.

The integral is zero because $x^3$ is odd: the area on $[-1, 0]$ is negative and exactly cancels the area on $[0, 1]$.

Common traps

Forgetting $+ C$ on an indefinite integral. Always include the constant of integration. It is a marked step.

Wrong division on linear inside argument. $\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C$, not $2 e^{2x}$. Divide by the coefficient, do not multiply.

Sign error on $\int \sin x \, dx$. The antiderivative is $-\cos x$, mirroring $\frac{d}{dx} \cos x = -\sin x$.

Reading off wrong limits. $\int_a^b f \, dx = F(b) - F(a)$, in that order. Reversing gives the negative.

Treating the integral as signed area for a negative function. $\int_0^{\pi} (-\sin x) \, dx$ counts the area below the axis as negative. If the geometric area is wanted, take absolute values or split the integral at the zeros.

Using the FTC over a discontinuity. $\int_{-1}^{1} \frac{1}{x^2} \, dx$ is improper because $\frac{1}{x^2}$ has a vertical asymptote at $x = 0$. Methods does not formally cover improper integrals; if QCAA asks an evaluation question, the integrand will be continuous on the interval.

In one sentence

Antidifferentiation reverses the standard derivatives (with linear inside arguments handled by dividing by the coefficient), and the Fundamental Theorem of Calculus evaluates the definite integral $\int_a^b f(x) \, dx$ as $F(b) - F(a)$ for any antiderivative $F$, with the definite integral itself defined as the limiting Riemann sum that geometrically gives signed area.