Calculus II — Integration

Question 1. (3 marks)

• An antiderivative of $f$ on an interval $I$ is a function $F$ such that $F'(x)=f(x)$ for all $x\in I$. (1)
• If $F$ and $G$ are both antiderivatives, then $(F-G)'=f-f=0$, so $F-G$ is constant (by the Mean Value Theorem / a function with zero derivative is constant). (1)
• Hence every antiderivative differs from $F$ by a constant, so the general form is $F(x)+C$. (1)

Question 2. (4 marks) (a) $\int(3x^2-4x+2/x)\,dx = x^3 - 2x^2 + 2\ln|x| + C$. (2) (1 for power terms, 1 for log term) (b) $\int(e^x + \sec^2 x)\,dx = e^x + \tan x + C$. (2)

Question 3. (5 marks)

• $u=x^2+1 \Rightarrow du = 2x\,dx \Rightarrow x\,dx = \tfrac12 du$. (1)
• Limits: $x=0\Rightarrow u=1$; $x=2\Rightarrow u=5$. (1)
• Integral $=\tfrac12\int_1^5 u^3\,du = \tfrac12\cdot\left[\tfrac{u^4}{4}\right]_1^5$. (1)
• $=\tfrac18(625-1)=\tfrac{624}{8}=78$. (2)

Question 4. (5 marks)

• By LIATE, let $u=x$ (algebraic), $dv=e^x dx$. Then $du=dx$, $v=e^x$. (2)
• $\int x e^x dx = xe^x - \int e^x dx$. (2)
• $= xe^x - e^x + C = e^x(x-1)+C$. (1)

Question 5. (5 marks)

• FTC Part 1: If $g(x)=\int_a^x f(t)\,dt$ with $f$ continuous, then $g'(x)=f(x)$. (1)
• FTC Part 2: If $f$ is continuous and $F$ is any antiderivative, then $\int_a^b f(x)\,dx = F(b)-F(a)$. (1)
• Let $G(x)=\int_1^{x^2}\sin t\,dt$. With $u=x^2$, $\frac{d}{dx}\int_1^{u}\sin t\,dt = \sin(u)\cdot\frac{du}{dx}$ (chain rule). (2)
• $=\sin(x^2)\cdot 2x = 2x\sin(x^2)$. (1)

Question 6. (4 marks)

• Odd power of sine: write $\sin^3 x = \sin x(1-\cos^2 x)$. (1)
• $\int \sin x (1-\cos^2 x)\cos^2 x\,dx$. Let $u=\cos x$, $du=-\sin x\,dx$. (1)
• $= -\int (1-u^2)u^2\,du = -\int(u^2-u^4)\,du = -\tfrac{u^3}{3}+\tfrac{u^5}{5}+C$. (1)
• $= \frac{\cos^5 x}{5} - \frac{\cos^3 x}{3} + C$. (1)

Question 7. (5 marks)

• Factor: $x^2-x-2=(x-2)(x+1)$. (1)
• $\frac{5x-4}{(x-2)(x+1)} = \frac{A}{x-2}+\frac{B}{x+1}$. Then $5x-4 = A(x+1)+B(x-2)$.
• $x=2$: $6=3A\Rightarrow A=2$. $x=-1$: $-9=-3B\Rightarrow B=3$. (2)
• Integral $=\int\left(\frac{2}{x-2}+\frac{3}{x+1}\right)dx = 2\ln|x-2|+3\ln|x+1|+C$. (2)

Question 8. (4 marks)

• $\int_1^\infty x^{-2}dx = \lim_{b\to\infty}\int_1^b x^{-2}dx$. (1)
• $=\lim_{b\to\infty}\left[-\tfrac1x\right]_1^b = \lim_{b\to\infty}\left(-\tfrac1b + 1\right)$. (2)
• $= 1$. Converges to $1$. (1)

Question 9. (5 marks)

• Intersections: $x^2=2x \Rightarrow x=0, 2$. (1)
• On $[0,2]$, $2x \ge x^2$, so area $=\int_0^2 (2x-x^2)\,dx$. (1)
• $=\left[x^2-\tfrac{x^3}{3}\right]_0^2 = 4-\tfrac83$. (2)
• $=\frac{4}{3}$. (1)

[
  {"claim":"Q3: definite integral equals 78","code":"x=symbols('x'); result = integrate(x*(x**2+1)**3,(x,0,2))==78"},
  {"claim":"Q7: partial fraction integral has coefficients 2 and 3","code":"x=symbols('x'); A,B=2,3; expr=simplify(A*(x+1)+B*(x-2)-(5*x-4)); result = expr==0"},
  {"claim":"Q8: improper integral converges to 1","code":"x=symbols('x'); result = integrate(1/x**2,(x,1,oo))==1"},
  {"claim":"Q9: area between curves equals 4/3","code":"x=symbols('x'); result = integrate(2*x-x**2,(x,0,2))==Rational(4,3)"}
]