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NSWMaths AdvancedQuick questions
Year 12: Calculus
Quick questions on Integration techniques: antiderivatives, substitution, definite integrals and the FTC
13short 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
Memorise these. The constant $C$ is omitted in definite integrals.
What is linear inside argument?Show answer
If the argument is linear, divide by the coefficient.
What is integration by substitution?Show answer
The substitution rule reverses the chain rule. To evaluate $\int f(g(x)) g'(x) \, dx$, set $u = g(x)$, so $du = g'(x) \, dx$. The integral becomes $\int f(u) \, du$, which you evaluate, then substitute back.
What is the Fundamental Theorem of Calculus?Show answer
If $F$ is any antiderivative of $f$ (so $F'(x) = f(x)$), then
What is standard antiderivative?Show answer
$\int (5 x^3 + 4 x - 7) \, dx = \frac{5 x^4}{4} + 2 x^2 - 7 x + C$.
What is substitution?Show answer
Evaluate $\int x \sqrt{x^2 + 4} \, dx$.
What is definite integral with substitution and changed limits?Show answer
Evaluate $\int_0^1 2 x e^{x^2} \, dx$.
What is using the FTC in reverse?Show answer
If $G(x) = \int_1^x \cos(t^2) \, dt$, then $G'(x) = \cos(x^2)$. No antiderivative needed.
What is forgetting the $+ C$?Show answer
An indefinite integral must include the constant of integration.
What is dividing by the wrong coefficient?Show answer
$\int e^{2 x} \, dx = \frac{1}{2} e^{2 x} + C$, not $2 e^{2 x} + C$.
What is substituting without changing $dx$?Show answer
When you substitute $u = g(x)$, the $dx$ must become $\frac{du}{g'(x)}$ (or the integrand must already contain a multiple of $g'(x) \, dx$).
What is mixing limits and the variable?Show answer
After a substitution, either change the limits to $u$ values or substitute back to $x$ before evaluating. Do not mix the two.
What is using the FTC with $|x|$ inside a logarithm wrongly?Show answer
$\int \frac{1}{x} \, dx = \ln |x| + C$. For a definite integral, the interval must not cross $x = 0$.