Calculus Term Questions

math-1109 MATH-1109 Differential and Integral Calculus

Subtopic 2.1: Basic Concepts of Integration

• What is meant by integration? — 2024
  (Definition question; no formula needed.)
• What is integration? Write the name of four applications of integration. — 2022
  (Definition + applications.)

Subtopic 2.2: Indefinite Integrals

• 2020 $\int \frac{dx}{(x+1)\sqrt{1+2x-x^2}}$
• 2020 $\int \frac{dx}{2\cos x+3\sin x}$
• 2021 $\int \frac{dx}{(2x+3)\sqrt{3x^2+3x+2}}$
• 2021 $\int \sqrt{3x-2}\sqrt{x^3-x+1}dx$
• 2021 $\int \frac{\sqrt{x\left(1+a\cos x\right)}}{1-\cos x}dx$
• 2023 $\int 2{\sin }^{-1}xdx$
• 2023 $\int \frac{dx}{5x^2-8x+7}$
• 2023 $\int x\ln (\sin x)dx$
• 2024 $\int \frac{2\cos x+3\sin x}{4\sin x-\cos x+5}dx$
• 2024 $\int \frac{x^2-1}{x^4+x^2+1}dx$
• 2024 $\int \frac{{\sin }^{-1}x}{\sqrt{1+x}}dx$
• 2024 Evaluate using differentiation under the integral sign:
  $\int \log (x+\sqrt{1+x^2})dx$
• 2024 $\int \frac{dx}{\sqrt{x^2+x-1}}$
• 2024 $\int \frac{dx}{3+3\cos x+4\sin x}$
• 2024 $\int \frac{dx}{\sqrt{x^2+2x+3}}$
• **2016 (a) $\int \frac{dx}{(x^2-1)\sqrt{x^2+1}}$ \quad
  (b) $\int \frac{xdx}{(x-1)(x^2+4)}$ \quad
  (c) $\int \frac{e^x}{2+\cos x-2\sin x}dx$ \quad
  (d) $\int \frac{dx}{x^4\sqrt{x^2-1}}$
• **2017 (a) $\int e^{2x}\sin (2x)dx$ \quad
  (b) $\int (x+2)\sqrt{x^2+x+1}dx$ \quad
  (c) $\int \frac{5\cos x+6}{2\cos x+\sin x+3}dx$ \quad
  (d) $\int \frac{dx}{(x^2-1)\sqrt{x^2-1}}$
• **2018 (a) $\int \frac{e^{{\tan }^{-1}x}}{1+x^2}dx$ \quad
  (b) $\int \frac{dx}{x^4+1}$ \quad
  (c) $\int x{\sin }^{2}xdx$ \quad
  (d) $\int \frac{1-\sin x+\cos x}{1+\sin x-\cos x}dx$
• 2018 $\int \frac{dx}{x\sqrt{1+x^3}}$
• **2019 ( (a) $\int \frac{dx}{(2x+1{)}^2\sqrt{3x^2+2}}$ \quad
  (b) $\int \frac{dx}{x^4+3x^2+1}$ \quad
  (c) $\int \frac{e^x\sqrt{x^2+1}}{(x+1{)}^2}dx$ \quad
  (d) $\int \frac{1-\sin x+\cos x}{1+\sin x-\cos x}dx$
• 2022
  (i) $\int \frac{\sin 2x-\sin x}{1+{\sin }^{2}x}dx$ \quad
  (ii) $\int \frac{1+x\ln x}{x}dx$ \quad
  (iii) $\int \frac{1-\cos x-\cos 2x}{1+\cos x-\cos 2x}dx$

Subtopic 2.3: Definite Integrals

• 2020 ${\int }_0^{\pi /2}\frac{\sin x}{1+{\cos }^{2}x}dx$
• 2020 ${\int }_0^{\pi /2}x{\sin }^{2}xdx$
• 2021, 2024 ${\int }_0^{\pi /2}\frac{x\sin x\cos x}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}dx$
• 2023 Write down some properties of definite integrals.
• 2023 ${\int }_0^{\pi /2}{\sin }^{5}x{\cos }^{4}xdx$
• 2023 ${\int }_0^{\pi /2}\frac{\sqrt{1-x}}{1+x}dx$
• 2024 ${\int }_0^1\frac{\ln \left(1+x^2\right)}{1+x^2}dx$
• 2024 (improper) ${\int }_0^{\infty }{e}^{-x^2}dx$
• 2024 Write down some general properties of definite integrals.
• 2024 ${\int }_1^2\sqrt{(x-1)(2-x)}dx$
• 2024 ${\int }_0^{\pi /2}\log (\sin x)dx$
• **2016 (a) ${\int }_0^{\pi /2}\frac{5\cos x+6}{2\cos x+\sin x+3}dx$ \quad
  (b) ${\int }_0^1\frac{dx}{(1+x)\sqrt{1+2x-x^2}}$ \quad
  (c) ${\int }_0^2(\sqrt{x}+\sqrt{2-x})dx$ \quad
  (d) $\int x{\cos }^{n}xdx$ (indefinite; included as asked)
• **2017
  (a) ${\int }_0^{\pi /2}{\sin }^6\theta {\cos }^3\theta d\theta$ \quad
  (b) ${\int }_0^{\pi }\frac{dx}{1-2a\cos x+a^2},0<a<1$ \quad
  (c) ${\int }_0^1\frac{\ln (1+x)}{1+x^2}dx$ \quad
  (d) ${\int }_0^1{x}^3(1-x^2)dx$
• 2018 (any three)
  (a) ${\int }_0^{\pi }\frac{dx}{1-2a\cos x+a^2}$ \quad
  (b) $\int \frac{x\tan x}{\sec x+\tan x}dx$ (indefinite) \quad
  (c) ${\int }_0^1\frac{\ln x}{\sqrt{1-x^2}}dx$ \quad
  (d) $\underset{n\to \infty }{\lim }\sum _{k=1}^n\frac{n^2}{n^3+k}$
• 2019 ${\int }_0^{\pi /2}x{\sin }^{6}x{\cos }^{4}xdx$
• 2019 ${\int }_0^1\frac{x^3{\sin }^{-1}x}{\sqrt{1-x^2}}dx$
• 2022 ${\int }_0^{\pi /2}\frac{dx}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}$
• 2022 Show that ${\int }_0^{\pi /2}\log \left(\sin \frac{x}{2}\right)dx=\log 2$ . (As noted, many texts use $\log (\sin x)$ here.)