Sections 1 to 7: substitution (indefinite and definite), integration by parts, repeated and tabular IBP, the cyclic trick, reduction formulas, and method selection第 1 至 7 节：换元法（不定积分与定积分）、分部积分法、多次及列表分部积分、循环积分技巧、递推公式与方法选择CALC II
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PART I · CORE TECHNIQUES核心技巧Computational fluency · 28 marks计算熟练度 · 28 分
Substitution and Integration by Parts换元法与分部积分法
Show all working. For substitution, state your choice of $u$ explicitly and change limits when the integral is definite. For integration by parts, state your choice of $u$ and $dv$ at each step. Leave answers in exact form; do not approximate.写出完整过程。换元时，须明确写出所选的 $u$，若为定积分须同步更换积分限。分部积分时，每步须写出所选的 $u$ 与 $dv$。答案保留精确形式，不得近似。
Q1MEDIUMCOREu-substitution: indefinite and definite换元法：不定积分与定积分[8 marks]
Evaluate each integral. In (b) and (c), change the limits of integration when you substitute.计算下列各积分。在 (b) 和 (c) 中，换元时须同步更换积分限。
(a)Apply integration by parts three times (using LIATE at each step) to evaluate $I$. Show each intermediate integral clearly.依次使用三次分部积分法（每步遵循 LIATE 规则）计算 $I$，清晰写出每一步的中间积分。[5]
(b)Alternatively, construct the tabular (column) integration table for $I$ and read off the result. Confirm that it matches part (a).另外，为 $I$ 构造列表积分表并读取结果，确认与 (a) 一致。[3]
Q4HARDCOREcyclic integral: the return trick循环积分：回归技巧[6 marks]
Let $\displaystyle I=\int e^{x}\sin x\,dx$.设 $\displaystyle I=\int e^{x}\sin x\,dx$。
(a)Apply integration by parts twice, keeping the same choice of which factor plays the role of $u$ at each step. Show that the original integral $I$ reappears on the right-hand side.使用两次分部积分法，每步保持相同因子担任 $u$ 的角色，证明原积分 $I$ 重新出现在等式右侧。[4]
(b)Hence solve the resulting equation for $I$.由此解方程，求出 $I$。[2]
PART II · DEFINITIONS AND PROOF定义与证明Rigorous arguments · 26 marks严格论证 · 26 分
Derivations and Reduction Formulas推导与递推公式
These items are graded on the logic of the derivation, not only the final answer. State every formula you invoke before using it. When deriving a reduction formula, you must show the IBP setup explicitly and algebraic rearrangement step by step.本部分按推导逻辑评分，而非仅看最终答案。使用任何公式前须先陈述该公式。推导递推公式时，须明确写出分部积分的设置及逐步代数变形。
Q5HARDPROOFderiving the IBP formula; reduction formula for $\int x^n e^x\,dx$推导分部积分公式；$\int x^n e^x\,dx$ 的递推公式[10 marks]
This question asks you to build the theory from the product rule.本题要求从乘积法则出发推导相关理论。
(a)Let $u$ and $v$ be differentiable functions of $x$. Starting from the product rule $(uv)'=u'v+uv'$, integrate both sides and rearrange to derive the integration by parts formula设 $u$ 与 $v$ 均为 $x$ 的可微函数。从乘积法则 $(uv)'=u'v+uv'$ 出发，对两边积分并整理，推导出分部积分公式
$$ \int u\,dv = uv - \int v\,du. $$
[3]
(b)Let $I_{n}=\int x^{n}e^{x}\,dx$ for $n\ge 1$. Using integration by parts with $u=x^{n}$ and $dv=e^{x}\,dx$, derive the reduction formula设 $I_{n}=\int x^{n}e^{x}\,dx$，$n\ge 1$。取 $u=x^{n}$，$dv=e^{x}\,dx$，利用分部积分法推导递推公式
$$ I_{n}=x^{n}e^{x}-n\,I_{n-1}. $$
[4]
(c)Apply the reduction formula from (b) repeatedly to evaluate $I_{3}=\int x^{3}e^{x}\,dx$, using $I_{0}=e^{x}+C$.反复应用 (b) 中的递推公式，利用 $I_{0}=e^{x}+C$ 计算 $I_{3}=\int x^{3}e^{x}\,dx$。[3]
Q6HARDPROOFWallis-type reduction formula for $\int \sin^n x\,dx$$\int \sin^n x\,dx$ 的 Wallis 型递推公式[10 marks]
Let $I_{n}=\displaystyle\int_{0}^{\pi/2}\sin^{n}x\,dx$ for integer $n\ge 2$.设 $I_{n}=\displaystyle\int_{0}^{\pi/2}\sin^{n}x\,dx$，$n$ 为整数且 $n\ge 2$。
(a)Write $\sin^{n}x=\sin^{n-1}x\cdot\sin x$ and apply integration by parts with $u=\sin^{n-1}x$, $dv=\sin x\,dx$. Show that after evaluating the boundary term at $x=0$ and $x=\tfrac{\pi}{2}$, you obtain将 $\sin^{n}x$ 写成 $\sin^{n-1}x\cdot\sin x$，取 $u=\sin^{n-1}x$，$dv=\sin x\,dx$，进行分部积分。证明在 $x=0$ 与 $x=\tfrac{\pi}{2}$ 处计算边界项后，得到
$$ I_{n} = \frac{n-1}{n}\,I_{n-2}. $$
[6]
(c)Use the reduction formula to compute $I_{5}=\displaystyle\int_{0}^{\pi/2}\sin^{5}x\,dx$. Show each step of the recursion.利用递推公式计算 $I_{5}=\displaystyle\int_{0}^{\pi/2}\sin^{5}x\,dx$，写出递推的每一步。[3]
Q7MEDIUMPROOFreduction formula for $\int x^n \ln x\,dx$$\int x^n \ln x\,dx$ 的递推公式[6 marks]
Let $J_{n}=\displaystyle\int x^{n}\ln x\,dx$ for integer $n\ge 0$, $x>0$.设 $J_{n}=\displaystyle\int x^{n}\ln x\,dx$，$n$ 为整数且 $n\ge 0$，$x>0$。
(a)Apply integration by parts once with a suitable choice of $u$ and $dv$, and show that选取适当的 $u$ 与 $dv$，进行一次分部积分，证明
$$ J_{n}=\frac{x^{n+1}\ln x}{n+1}-\frac{x^{n+1}}{(n+1)^{2}}+C. $$
[4]
(b)Verify your answer in (a) by differentiating the right-hand side and confirming you recover the integrand $x^{n}\ln x$.对 (a) 结果右侧求导，验证可还原被积函数 $x^{n}\ln x$，从而确认答案正确。[2]
PART III · APPLICATIONS AND SYNTHESIS应用与综合Extended problems · 28 marks综合题 · 28 分
Definite Integrals, Area, and Method Selection定积分、面积与方法选择
Set up each problem clearly before computing. A substitution in a definite integral must be accompanied by a change of limits. In Q10, justify your method choice before computing.计算前须清晰建立每道题的设置。定积分中换元必须同步更换积分限。在 Q10 中，计算前须先说明方法选择的依据。
Q8HARDAPPLIEDdefinite integral requiring substitution then integration by parts先换元后分部积分的定积分[8 marks]
(a)Let $u=x^{2}+1$. Show that the integral transforms to令 $u=x^{2}+1$，证明积分变形为
$$ \frac{1}{2}\int_{1}^{2}(u-1)\sqrt{u}\,du. $$
State the new limits and express $x^{3}\,dx$ in terms of $u$ and $du$.写出新的积分限，并将 $x^{3}\,dx$ 用 $u$ 和 $du$ 表示。[3]
(b)Expand $(u-1)\sqrt{u}$ and evaluate the resulting integral using the power rule.展开 $(u-1)\sqrt{u}$，利用幂函数积分法则计算所得积分。[3]
(c)Hence state the exact value of the original definite integral.由此写出原定积分的精确值。[2]
Q9HARDAPPLIEDarea under a curve via integration by parts利用分部积分法求曲线下面积[10 marks]
Let $f(x)=x\ln x$ on the interval $[1,e]$.设 $f(x)=x\ln x$ 定义在区间 $[1,e]$ 上。
(a)Find $f'(x)$ and determine whether $f$ is increasing or decreasing on $(1,e)$.求 $f'(x)$，判断 $f$ 在 $(1,e)$ 上是单调递增还是单调递减。[2]
(b)Evaluate the definite integral $\displaystyle\int_{1}^{e}x\ln x\,dx$ using integration by parts. Show the $uv$ boundary evaluation and the remaining integral clearly.利用分部积分法计算定积分 $\displaystyle\int_{1}^{e}x\ln x\,dx$，清晰写出 $uv$ 边界值的计算及余下积分。[5]
(c)Hence find the exact area of the region bounded by $y=x\ln x$, the $x$-axis, and the vertical lines $x=1$ and $x=e$. Justify that $f(x)\ge 0$ on $[1,e]$ before stating the area.由此求由 $y=x\ln x$、$x$ 轴以及直线 $x=1$ 与 $x=e$ 围成区域的精确面积。写出面积前须先说明 $f(x)\ge 0$ 在 $[1,e]$ 上成立的理由。[3]
Q10HARDAPPLIEDmethod selection: substitution vs. IBP vs. direct方法选择：换元法、分部积分法与直接法[10 marks]
For each integral below, name the method you will use (substitution, IBP, direct antiderivative, or other), state the key step that makes the method work, then evaluate.对下列各积分，写出所选方法（换元法、分部积分法、直接求原函数或其他），说明使该方法有效的关键步骤，然后求值。