Covers Lessons 1–5: antiderivatives, power rule, exponentials, logarithms, definite integrals, and the Fundamental Theorem of Calculus.
Assessment
Select the best answer for each question.
$\int x^3 \, dx$ equals:
$\int e^{2x} \, dx$ equals:
$\int \frac{1}{x} \, dx$ equals:
$\int_0^2 x^2 \, dx$ equals:
$\frac{d}{dx}\left(\int_0^x t^3 \, dt\right)$ equals:
$\int \sqrt{x} \, dx$ equals:
$\int_1^e \frac{1}{x} \, dx$ equals:
$\int \frac{1}{x^3} \, dx$ equals:
Short Answer
Find $\int (2e^{3x} - \frac{4}{x} + x^2) \, dx$.
Evaluate $\int_0^2 (3x^2 + 2x + 1) \, dx$.
Find $\frac{d}{dx}\left(\int_1^{x^2} (t + 1) \, dt\right)$.
Q1: C — Power rule: $\frac{x^4}{4} + C$.
Q2: B — $\frac{1}{2}e^{2x} + C$.
Q3: B — $\ln|x| + C$.
Q4: C — $[\frac{x^3}{3}]_0^2 = \frac{8}{3}$.
Q5: B — FTC Part 1: $x^3$.
Q6: C — $\int x^{1/2} \, dx = \frac{2}{3}x^{3/2} + C$.
Q7: B — $[\ln x]_1^e = 1 - 0 = 1$.
Q8: B — $\int x^{-3} \, dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$.
Q9 (3 marks): $\frac{2}{3}e^{3x} - 4\ln|x| + \frac{x^3}{3} + C$ (1 mark per correct term).
Q10 (3 marks): $F(x) = x^3 + x^2 + x$ [1]. $[F(x)]_0^2 = (8 + 4 + 2) - 0 = 14$ [2].
Q11 (3 marks): By FTC Part 1 with chain rule: $(x^2 + 1) \cdot 2x$ [2] $= 2x^3 + 2x$ [1].