Back to the full dot-point answer
QLDMath MethodsQuick questions
Unit 3: Further calculus and statistics
Quick questions on Antiderivatives and the Fundamental Theorem of Calculus (QCE Mathematical Methods Unit 3)
15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is standard antiderivatives?Show answer
The constant $C$ is required on every indefinite integral.
What is linear inside argument?Show answer
If the argument is linear ($ax + b$), divide by the coefficient of $x$.
What is the Fundamental Theorem of Calculus?Show answer
If $F$ is any antiderivative of $f$ (so $F'(x) = f(x)$), then
What is the Riemann sum definition?Show answer
The definite integral $\int_a^b f(x) \, dx$ is defined as the limit of a Riemann sum:
What is properties of the definite integral?Show answer
These properties speed up Paper 1 evaluation.
What is standard indefinite integral?Show answer
$\int (4 x^3 - 6 x + 2) \, dx = x^4 - 3 x^2 + 2 x + C.$
What is exponential with linear argument?Show answer
$\int e^{-3 x} \, dx = \dfrac{e^{-3 x}}{-3} + C = -\dfrac{1}{3} e^{-3 x} + C.$
What is definite integral?Show answer
Evaluate $\int_1^3 (x^2 + 2 x) \, dx$.
What is definite integral with the FTC reverse?Show answer
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.
What is negative area?Show answer
Evaluate $\int_{-1}^{1} x^3 \, dx$.
What is forgetting $+ C$ on an indefinite integral?Show answer
Always include the constant of integration. It is a marked step.
What is wrong division on linear inside argument?Show answer
$\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C$, not $2 e^{2x}$. Divide by the coefficient, do not multiply.
What is sign error on $\int \sin x \, dx$?Show answer
The antiderivative is $-\cos x$, mirroring $\frac{d}{dx} \cos x = -\sin x$.
What is reading off wrong limits?Show answer
$\int_a^b f \, dx = F(b) - F(a)$, in that order. Reversing gives the negative.
What is treating the integral as signed area for a negative function?Show answer
$\int_0^{\pi} (-\sin x) \, dx$ counts the area below the axis as negative. If the geometric area is wanted, take absolute values or split the integral at the zeros.