Calculus II — Master Practice Exam

Core Concepts & Formulas

Review each topic before tackling the practice questions

C-01 Integration by Parts ▾

$$\int u\,dv = uv - \int v\,du$$

LIATE Rule for choosing $u$: Logarithms → Inverse trig → Algebraic → Trig → Exponential.

Example

Evaluate $\int x e^x\,dx$.

Let $u=x$, $dv=e^x dx$. Then $du=dx$, $v=e^x$.
$\displaystyle = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1)+C$

C-02 Trigonometric Integrals ▾

$$\sin^2 x = \tfrac{1-\cos 2x}{2},\quad \cos^2 x = \tfrac{1+\cos 2x}{2}$$

For $\int \sin^m x\cos^n x\,dx$: if $m$ is odd, save one $\sin x$ and convert the rest using $\sin^2 x=1-\cos^2 x$. If $n$ is odd, analogous with $\cos x$. Both even: use half-angle identities.

Example

Evaluate $\int \sin^2 x\,dx = \int \tfrac{1-\cos 2x}{2}\,dx = \tfrac{x}{2}-\tfrac{\sin 2x}{4}+C$

C-03 Trigonometric Substitution ▾

$\sqrt{a^2-x^2}$: let $x=a\sin\theta$
$\sqrt{a^2+x^2}$: let $x=a\tan\theta$
$\sqrt{x^2-a^2}$: let $x=a\sec\theta$

Example

Evaluate $\displaystyle\int\frac{dx}{\sqrt{4-x^2}}$. Let $x=2\sin\theta$, $dx=2\cos\theta\,d\theta$.
$\displaystyle =\int\frac{2\cos\theta\,d\theta}{2\cos\theta}=\theta+C=\arcsin\!\left(\tfrac{x}{2}\right)+C$

C-04 Partial Fractions ▾

Decompose a proper rational function into simpler fractions before integrating. For a distinct linear factor $(x-a)$, include $\dfrac{A}{x-a}$. For a repeated factor $(x-a)^2$, include $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$.

$$\frac{1}{x^2-1}=\frac{1}{(x-1)(x+1)}=\frac{1/2}{x-1}-\frac{1/2}{x+1}$$

Example

$\displaystyle\int\frac{dx}{x^2-1}=\frac{1}{2}\ln|x-1|-\frac{1}{2}\ln|x+1|+C=\frac{1}{2}\ln\left|\frac{x-1}{x+1}\right|+C$

C-05 Improper Integrals ▾

$$\int_a^\infty f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx$$

p-test: $\displaystyle\int_1^\infty \frac{dx}{x^p}$ converges if and only if $p>1$.

Example

$\displaystyle\int_1^\infty \frac{dx}{x^2}=\lim_{b\to\infty}\!\left[-\frac{1}{x}\right]_1^b=0-(-1)=1$ (converges)

C-06 Sequences & Series Convergence ▾

Key Tests:

Ratio Test: $L=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$. Converges if $L<1$, diverges if $L>1$.

Integral Test: $\sum a_n$ converges iff $\int_1^\infty f(x)\,dx$ converges.

Alternating Series: $\sum(-1)^n b_n$ converges if $b_n\searrow 0$.

C-07 Power Series & Radius of Convergence ▾

$$\sum_{n=0}^\infty c_n(x-a)^n \quad R=\frac{1}{\limsup_{n\to\infty}|c_n|^{1/n}}$$

Use the Ratio Test: $R=\lim_{n\to\infty}\left|\dfrac{c_n}{c_{n+1}}\right|$. The series converges absolutely for $|x-a|<R$.

C-08 Taylor & Maclaurin Series ▾

$$f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Key Maclaurin series to memorize:

$e^x=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}$

$\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$\cos x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n x^{2n}}{(2n)!}$

$\dfrac{1}{1-x}=\displaystyle\sum_{n=0}^\infty x^n,\quad |x|<1$

C-09 Polar Coordinates & Area ▾

$$A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$$

Arc length in polar: $\displaystyle L=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$

Example

Area inside $r=\cos\theta$ (a circle of radius $\frac{1}{2}$):
$\displaystyle A=\frac{1}{2}\int_0^\pi \cos^2\theta\,d\theta=\frac{\pi}{4}$

C-10 Parametric Equations ▾

$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$

Arc length: $L=\displaystyle\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$

Example

If $x=t^2,\,y=t^3$, then $\dfrac{dy}{dx}=\dfrac{3t^2}{2t}=\dfrac{3t}{2}$.

Practice Exam

20 questions · Select the best answer for each

1 Integration by Parts

Evaluate $\displaystyle\int x\cos x\,dx$.

2 Trigonometric Integrals

Evaluate $\displaystyle\int_0^{\pi/2}\sin^2 x\,dx$.

3 Trigonometric Substitution

Which substitution is most appropriate to evaluate $\displaystyle\int\frac{x^2\,dx}{\sqrt{9-x^2}}$?

4 Partial Fractions

Evaluate $\displaystyle\int\frac{3x+1}{x^2+x}\,dx$.

5 Improper Integrals

Determine whether $\displaystyle\int_1^\infty \frac{dx}{\sqrt{x}}$ converges or diverges. If it converges, give its value.

6 Series — Ratio Test

Apply the Ratio Test to $\displaystyle\sum_{n=1}^\infty \frac{n!}{3^n}$. What is the conclusion?

7 Alternating Series

Which series converges by the Alternating Series Test?

8 Power Series — Radius of Convergence

Find the radius of convergence of $\displaystyle\sum_{n=0}^\infty \frac{x^n}{n+1}$.

9 Taylor Series

What is the Maclaurin series for $f(x)=e^{-x^2}$?

10 Polar Area

Find the area enclosed by the cardioid $r = 1+\cos\theta$.

11 Parametric Slope

A curve is given parametrically by $x=t^2-1$, $y=t^3-3t$. At which value(s) of $t$ does the curve have a horizontal tangent?

12 Volume of Revolution — Disk Method

Find the volume of the solid generated by revolving the region bounded by $y=\sqrt{x}$, $x=4$, and the $x$-axis about the $x$-axis.

13 Arc Length

Set up (do not evaluate) the arc length of $y=x^{3/2}$ from $x=0$ to $x=4$.

14 Geometric Series

Find the sum of the series $\displaystyle\sum_{n=0}^\infty \left(\frac{2}{3}\right)^n$.

15 Comparison Test

Use the Limit Comparison Test to determine convergence of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2+5n+4}$.

16 Integration by Parts — Reduction

Evaluate $\displaystyle\int \ln x\,dx$.

17 p-Series Test

Which of the following series diverges?

18 Taylor Polynomial Error

The third-degree Taylor polynomial for $f(x)=\sin x$ centered at $x=0$ is $P_3(x)=x-\dfrac{x^3}{6}$. The Lagrange error bound for $|P_3(x)-\sin x|$ when $|x|\le 0.5$ uses which maximum derivative value?

19 Surface Area of Revolution

The formula for the surface area generated by revolving $y=f(x)$, $a\le x\le b$, about the $x$-axis is:

20 Integral Test / p-Series Application

Which statement about $\displaystyle\int_1^\infty \frac{\ln x}{x^2}\,dx$ is correct?

🎓 Exam Complete!

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Correct

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Wrong

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Time

Complete Solutions & Explanations

Step-by-step solutions for all 20 questions

1 Integration by Parts

Correct Answer: B — $x\sin x+\cos x+C$

Let $u=x$, $dv=\cos x\,dx$. Then $du=dx$, $v=\sin x$.

$$\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x +\cos x + C$$

Differentiating to verify: $\frac{d}{dx}(x\sin x+\cos x)=\sin x+x\cos x-\sin x=x\cos x\checkmark$

2 Trigonometric Integrals

Correct Answer: B — $\dfrac{\pi}{4}$

Use the half-angle identity $\sin^2 x=\dfrac{1-\cos 2x}{2}$:

$$\int_0^{\pi/2}\sin^2 x\,dx=\int_0^{\pi/2}\frac{1-\cos 2x}{2}\,dx=\left[\frac{x}{2}-\frac{\sin 2x}{4}\right]_0^{\pi/2}$$

$$=\frac{\pi/2}{2}-\frac{\sin\pi}{4}-0=\frac{\pi}{4}-0=\frac{\pi}{4}$$

3 Trigonometric Substitution

Correct Answer: B — $x=3\sin\theta$

The integrand contains $\sqrt{9-x^2}=\sqrt{a^2-x^2}$ with $a=3$. The standard substitution for this form is $x=a\sin\theta$, i.e., $x=3\sin\theta$.

This converts $\sqrt{9-x^2}=\sqrt{9-9\sin^2\theta}=3|\cos\theta|$.

4 Partial Fractions

Correct Answer: A — $\ln|x|+2\ln|x+1|+C$

Factor: $x^2+x=x(x+1)$. Write $\dfrac{3x+1}{x(x+1)}=\dfrac{A}{x}+\dfrac{B}{x+1}$.

Multiply both sides by $x(x+1)$: $3x+1=A(x+1)+Bx$.

Set $x=0$: $1=A$. Set $x=-1$: $-2=-B$, so $B=2$.

$$\int\frac{3x+1}{x(x+1)}\,dx=\int\frac{1}{x}\,dx+\int\frac{2}{x+1}\,dx=\ln|x|+2\ln|x+1|+C$$

5 Improper Integrals

Correct Answer: C — Diverges

This is a $p$-integral with $p=\frac{1}{2} < 1$, so it diverges. Verify directly:

$$\int_1^\infty x^{-1/2}\,dx=\lim_{b\to\infty}\left[2\sqrt{x}\right]_1^b=\lim_{b\to\infty}(2\sqrt{b}-2)=+\infty$$

6 Ratio Test

Correct Answer: B — Diverges, since $L=\infty>1$

$$L=\lim_{n\to\infty}\left|\frac{(n+1)!/3^{n+1}}{n!/3^n}\right|=\lim_{n\to\infty}\frac{n+1}{3}=+\infty>1$$

Since $L>1$, the series diverges by the Ratio Test.

7 Alternating Series Test

Correct Answer: B — $\sum(-1)^n\dfrac{1}{n^2}$

The Alternating Series Test requires (1) terms $b_n$ are decreasing and (2) $\lim_{n\to\infty}b_n=0$.

For option B: $b_n=1/n^2$ is decreasing and $\to 0$. ✓

Option A fails: $b_n=n/(n+1)\to 1\ne 0$. Options C & D fail similarly.

8 Radius of Convergence

Correct Answer: C — $R=1$

Apply the Ratio Test with $a_n=\dfrac{x^n}{n+1}$:

$$L=\lim_{n\to\infty}\left|\frac{x^{n+1}/(n+2)}{x^n/(n+1)}\right|=|x|\lim_{n\to\infty}\frac{n+1}{n+2}=|x|$$

Converges when $L=|x|<1$, so $R=1$.

9 Maclaurin Series

Correct Answer: A — $\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^n x^{2n}}{n!}$

Start from $e^u=\displaystyle\sum_{n=0}^\infty\dfrac{u^n}{n!}$. Substitute $u=-x^2$:

$$e^{-x^2}=\sum_{n=0}^\infty\frac{(-x^2)^n}{n!}=\sum_{n=0}^\infty\frac{(-1)^n x^{2n}}{n!}$$

10 Polar Area

Correct Answer: A — $\dfrac{3\pi}{2}$

$$A=\frac{1}{2}\int_0^{2\pi}(1+\cos\theta)^2\,d\theta=\frac{1}{2}\int_0^{2\pi}(1+2\cos\theta+\cos^2\theta)\,d\theta$$

Over $[0,2\pi]$: $\int_0^{2\pi}1\,d\theta=2\pi$, $\int_0^{2\pi}2\cos\theta\,d\theta=0$, $\int_0^{2\pi}\cos^2\theta\,d\theta=\pi$.

$$A=\frac{1}{2}(2\pi+0+\pi)=\frac{3\pi}{2}$$

11 Parametric Slope

Correct Answer: B — $t=\pm 1$

Horizontal tangent when $dy/dt=0$ and $dx/dt\ne 0$.

$$\frac{dy}{dt}=3t^2-3=3(t^2-1)=0\implies t=\pm 1$$

Check $dx/dt=2t$: at $t=\pm1$, $dx/dt=\pm2\ne0$. Both are valid horizontal tangent points.

12 Volume — Disk Method

Correct Answer: A — $8\pi$

$$V=\pi\int_0^4 (\sqrt{x})^2\,dx=\pi\int_0^4 x\,dx=\pi\left[\frac{x^2}{2}\right]_0^4=\pi\cdot 8=8\pi$$

13 Arc Length

Correct Answer: A — $\displaystyle\int_0^4\sqrt{1+\dfrac{9x}{4}}\,dx$

Arc length formula: $L=\int_a^b\sqrt{1+[y']^2}\,dx$. Compute $y'=\dfrac{3}{2}x^{1/2}$.

$$[y']^2=\frac{9}{4}x$$

$$L=\int_0^4\sqrt{1+\frac{9x}{4}}\,dx$$

14 Geometric Series

Correct Answer: B — $3$

This is a geometric series with first term $a=1$ and common ratio $r=\dfrac{2}{3}$. Since $|r|<1$:

$$S=\frac{a}{1-r}=\frac{1}{1-2/3}=\frac{1}{1/3}=3$$

15 Comparison Test

Correct Answer: B — Converges by comparison to $\sum 1/n^2$

Compare with $b_n=1/n^2$:

$$\lim_{n\to\infty}\frac{1/(n^2+5n+4)}{1/n^2}=\lim_{n\to\infty}\frac{n^2}{n^2+5n+4}=1>0$$

Since the limit is a finite positive number and $\sum 1/n^2$ converges ($p=2>1$), the original series converges.

16 Integration by Parts

Correct Answer: B — $x\ln x - x + C$

Let $u=\ln x$, $dv=dx$. Then $du=\dfrac{1}{x}dx$, $v=x$.

$$\int\ln x\,dx=x\ln x-\int x\cdot\frac{1}{x}\,dx=x\ln x-\int 1\,dx=x\ln x-x+C$$

17 p-Series Test

Correct Answer: C — $\sum 1/\sqrt{n}$ diverges

This is a $p$-series with $p=\frac{1}{2}\le 1$, so it diverges. The others have $p=2,\,\frac{3}{2},\,1.01$ — all greater than 1, so they converge.

18 Taylor Error Bound

Correct Answer: A

The Lagrange remainder for a degree-$n$ polynomial is $\left|\dfrac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\right|$. Here $n=3$, so the bound uses $f^{(4)}$.

Since $f(x)=\sin x$, all derivatives are bounded by 1. So the error bound is:

$$|E|\le\frac{|f^{(4)}(c)|}{4!}|x|^4\le\frac{1\cdot(0.5)^4}{4!}=\frac{(0.5)^4}{24}$$

19 Surface Area Formula

Correct Answer: B

$$S=2\pi\int_a^b f(x)\sqrt{1+[f'(x)]^2}\,dx$$

This formula comes from wrapping the arc-length element $ds=\sqrt{1+[f']^2}\,dx$ around a circle of radius $f(x)$.

Note: Option C is the volume (Disk Method), not surface area.

20 Improper Integral — IBP

Correct Answer: B — Converges to $1$

Use integration by parts: let $u=\ln x$, $dv=x^{-2}dx$. Then $du=\frac{1}{x}dx$, $v=-x^{-1}$.

$$\int_1^b\frac{\ln x}{x^2}\,dx=\left[-\frac{\ln x}{x}\right]_1^b+\int_1^b\frac{1}{x^2}\,dx=-\frac{\ln b}{b}+0+\left[-\frac{1}{x}\right]_1^b$$

$$=-\frac{\ln b}{b}-\frac{1}{b}+1$$

As $b\to\infty$: $\ln b/b\to 0$ (L'Hôpital) and $1/b\to 0$. So the integral converges to $1$.