Practice exam · Calculus II · Units 3B–4B

Calculus II Practice Final Exam with Complete Solutions

A balanced twenty-five-question Calculus II final with study map and complete worked key.

25 problems120 min estimated timeCumulative progression

What is included

Prepare for a Calculus II final with integration methods, applications, convergence, power series, and Taylor series.

Skills assessed

method selection
convergence proof
series construction
course synthesis

Prerequisites

Calculus II Units 3B through 4B

Exam conditions

Suggested time: 120 minutes

Points: 4 points per question; 100 points total

Calculator: Scientific calculator permitted; computer algebra is not required.

This is an original BetterGrades practice exam, not a released institutional exam.

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF

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Evaluate $\int xe^x dx$ .
Evaluate $\int\sin^3x\cos xdx$ .
Evaluate $\int1/(x^2-1)dx$ .
Determine convergence of $\int_1^\infty x^{-2}dx$ .
Find area between y=x and y=x² on [0,1].
Rotate y=x on [0,2] about x-axis. Find volume.
A force F(x)=4x acts from x=0 to 3. Find work.
Find $\lim n/(n+1)$ .
Sum $\sum_{n=0}^\infty3(1/2)^n$ .
Test $\sum n/(n+1)$ for convergence.
Test $\sum1/n^{3/2}$ .
Test $\sum1/(n^2+4)$ .
Test $\sum(3n+1)/(n^2+2)$ .
Test $\sum n!/4^n$ .
Test $\sum(2n/(3n+1))^n$ .
Test $\sum(-1)^{n-1}/n$ .
Classify $\sum(-1)^n/n^2$ .
Find R for $\sum x^n/5^n$ .
Find interval for $\sum(x-1)^n/n$ .
Write the Maclaurin series for cos x.
Find T_3 for e^x at 0.
Bound alternating-series error after four terms of $\sum(-1)^{n-1}/n$ .
For x=t²,y=t³ find dy/dx.
Convert r=2cosθ to Cartesian form.
Why must power-series endpoints be tested separately?

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Evaluate $\int xe^x dx$ .

Answer: $e^x(x-1)+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use integration by parts.
The verified result is $e^x(x-1)+C$ .

Problem 2: Evaluate $\int\sin^3x\cos xdx$ .

Answer: $\sin^4x/4+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use u=sin x.
The verified result is $\sin^4x/4+C$ .

Problem 3: Evaluate $\int1/(x^2-1)dx$ .

Answer: $\frac12\ln|\frac{x-1}{x+1}|+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use partial fractions.
The verified result is $\frac12\ln|\frac{x-1}{x+1}|+C$ .

Problem 4: Determine convergence of $\int_1^\infty x^{-2}dx$ .

Answer: $converges to 1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Evaluate the defining limit.
The verified result is $converges to 1$ .

Problem 5: Find area between y=x and y=x² on [0,1].

Answer: $1/6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate top minus bottom.
The verified result is $1/6$ .

Problem 6: Rotate y=x on [0,2] about x-axis. Find volume.

Answer: $8\pi/3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use disks: π∫x²dx.
The verified result is $8\pi/3$ .

Problem 7: A force F(x)=4x acts from x=0 to 3. Find work.

Answer: $18$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate force over displacement.
The verified result is $18$ .

Problem 8: Find $\lim n/(n+1)$ .

Answer: $1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide by n.
The verified result is $1$ .

Problem 9: Sum $\sum_{n=0}^\infty3(1/2)^n$ .

Answer: $6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use a/(1-r).
The verified result is $6$ .

Problem 10: Test $\sum n/(n+1)$ for convergence.

Answer: $diverges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Terms do not approach zero.
The verified result is $diverges$ .

Problem 11: Test $\sum1/n^{3/2}$ .

Answer: $converges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

It is a p-series with p>1.
The verified result is $converges$ .

Problem 12: Test $\sum1/(n^2+4)$ .

Answer: $converges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Compare with 1/n².
The verified result is $converges$ .

Problem 13: Test $\sum(3n+1)/(n^2+2)$ .

Answer: $diverges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Limit compare with 1/n.
The verified result is $diverges$ .

Problem 14: Test $\sum n!/4^n$ .

Answer: $diverges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The ratio (n+1)/4 eventually exceeds 1.
The verified result is $diverges$ .

Problem 15: Test $\sum(2n/(3n+1))^n$ .

Answer: $converges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Root-test limit is 2/3.
The verified result is $converges$ .

Problem 16: Test $\sum(-1)^{n-1}/n$ .

Answer: $converges conditionally$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

AST applies; harmonic absolute series diverges.
The verified result is $converges conditionally$ .

Problem 17: Classify $\sum(-1)^n/n^2$ .

Answer: $converges absolutely$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The absolute series is a p-series with p=2.
The verified result is $converges absolutely$ .

Problem 18: Find R for $\sum x^n/5^n$ .

Answer: $R=5$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

It is geometric with ratio x/5.
The verified result is $R=5$ .

Problem 19: Find interval for $\sum(x-1)^n/n$ .

Answer: $[0,2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

R=1, then test both endpoints.
The verified result is $[0,2)$ .

Problem 20: Write the Maclaurin series for cos x.

Answer: $\sum(-1)^nx^{2n}/(2n)!$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the derivative cycle.
The verified result is $\sum(-1)^nx^{2n}/(2n)!$ .

Problem 21: Find T_3 for e^x at 0.

Answer: $1+x+x^2/2+x^3/6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use derivatives of e^x.
The verified result is $1+x+x^2/2+x^3/6$ .

Problem 22: Bound alternating-series error after four terms of $\sum(-1)^{n-1}/n$ .

Answer: $\le1/5$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the first omitted term.
The verified result is $\le1/5$ .

Problem 23: For x=t²,y=t³ find dy/dx.

Answer: $3t/2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide dy/dt by dx/dt.
The verified result is $3t/2$ .

Problem 24: Convert r=2cosθ to Cartesian form.

Answer: $(x-1)^2+y^2=1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use r²=2r cosθ.
The verified result is $(x-1)^2+y^2=1$ .

Problem 25: Why must power-series endpoints be tested separately?

Answer: $The ratio/root test is inconclusive when |x-a|=R.$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

At the boundary the limiting ratio equals 1.
The verified result is $The ratio/root test is inconclusive when |x-a|=R.$ .

Common errors

Choosing an integration or convergence test from surface appearance alone.
Skipping endpoint checks.
Claiming a Taylor representation without a convergence or remainder argument.

Practice exam · Calculus II · Units 3B–4B

Calculus II Practice Final Exam with Complete Solutions

A balanced twenty-five-question Calculus II final with study map and complete worked key.

25 problems120 min estimated timeCumulative progression

What is included

Prepare for a Calculus II final with integration methods, applications, convergence, power series, and Taylor series.

Skills assessed

method selection
convergence proof
series construction
course synthesis

Prerequisites

Calculus II Units 3B through 4B

Exam conditions

Suggested time: 120 minutes

Points: 4 points per question; 100 points total

Calculator: Scientific calculator permitted; computer algebra is not required.

This is an original BetterGrades practice exam, not a released institutional exam.

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF

Device-only study control

Ready for an honest first attempt?

This control sends only the resource id and type—never your answers or identity.

Printable preview

Evaluate $\int xe^x dx$ .
Evaluate $\int\sin^3x\cos xdx$ .
Evaluate $\int1/(x^2-1)dx$ .
Determine convergence of $\int_1^\infty x^{-2}dx$ .
Find area between y=x and y=x² on [0,1].
Rotate y=x on [0,2] about x-axis. Find volume.
A force F(x)=4x acts from x=0 to 3. Find work.
Find $\lim n/(n+1)$ .
Sum $\sum_{n=0}^\infty3(1/2)^n$ .
Test $\sum n/(n+1)$ for convergence.
Test $\sum1/n^{3/2}$ .
Test $\sum1/(n^2+4)$ .
Test $\sum(3n+1)/(n^2+2)$ .
Test $\sum n!/4^n$ .
Test $\sum(2n/(3n+1))^n$ .
Test $\sum(-1)^{n-1}/n$ .
Classify $\sum(-1)^n/n^2$ .
Find R for $\sum x^n/5^n$ .
Find interval for $\sum(x-1)^n/n$ .
Write the Maclaurin series for cos x.
Find T_3 for e^x at 0.
Bound alternating-series error after four terms of $\sum(-1)^{n-1}/n$ .
For x=t²,y=t³ find dy/dx.
Convert r=2cosθ to Cartesian form.
Why must power-series endpoints be tested separately?

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Evaluate $\int xe^x dx$ .

Answer: $e^x(x-1)+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use integration by parts.
The verified result is $e^x(x-1)+C$ .

Problem 2: Evaluate $\int\sin^3x\cos xdx$ .

Answer: $\sin^4x/4+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use u=sin x.
The verified result is $\sin^4x/4+C$ .

Problem 3: Evaluate $\int1/(x^2-1)dx$ .

Answer: $\frac12\ln|\frac{x-1}{x+1}|+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use partial fractions.
The verified result is $\frac12\ln|\frac{x-1}{x+1}|+C$ .

Problem 4: Determine convergence of $\int_1^\infty x^{-2}dx$ .

Answer: $converges to 1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Evaluate the defining limit.
The verified result is $converges to 1$ .

Problem 5: Find area between y=x and y=x² on [0,1].

Answer: $1/6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate top minus bottom.
The verified result is $1/6$ .

Problem 6: Rotate y=x on [0,2] about x-axis. Find volume.

Answer: $8\pi/3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use disks: π∫x²dx.
The verified result is $8\pi/3$ .

Problem 7: A force F(x)=4x acts from x=0 to 3. Find work.

Answer: $18$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate force over displacement.
The verified result is $18$ .

Problem 8: Find $\lim n/(n+1)$ .

Answer: $1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide by n.
The verified result is $1$ .

Problem 9: Sum $\sum_{n=0}^\infty3(1/2)^n$ .

Answer: $6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use a/(1-r).
The verified result is $6$ .

Problem 10: Test $\sum n/(n+1)$ for convergence.

Answer: $diverges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Terms do not approach zero.
The verified result is $diverges$ .

Problem 11: Test $\sum1/n^{3/2}$ .

Answer: $converges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

It is a p-series with p>1.
The verified result is $converges$ .

Problem 12: Test $\sum1/(n^2+4)$ .

Answer: $converges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Compare with 1/n².
The verified result is $converges$ .

Problem 13: Test $\sum(3n+1)/(n^2+2)$ .

Answer: $diverges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Limit compare with 1/n.
The verified result is $diverges$ .

Problem 14: Test $\sum n!/4^n$ .

Answer: $diverges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The ratio (n+1)/4 eventually exceeds 1.
The verified result is $diverges$ .

Problem 15: Test $\sum(2n/(3n+1))^n$ .

Answer: $converges$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Root-test limit is 2/3.
The verified result is $converges$ .

Problem 16: Test $\sum(-1)^{n-1}/n$ .

Answer: $converges conditionally$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

AST applies; harmonic absolute series diverges.
The verified result is $converges conditionally$ .

Problem 17: Classify $\sum(-1)^n/n^2$ .

Answer: $converges absolutely$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The absolute series is a p-series with p=2.
The verified result is $converges absolutely$ .

Problem 18: Find R for $\sum x^n/5^n$ .

Answer: $R=5$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

It is geometric with ratio x/5.
The verified result is $R=5$ .

Problem 19: Find interval for $\sum(x-1)^n/n$ .

Answer: $[0,2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

R=1, then test both endpoints.
The verified result is $[0,2)$ .

Problem 20: Write the Maclaurin series for cos x.

Answer: $\sum(-1)^nx^{2n}/(2n)!$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the derivative cycle.
The verified result is $\sum(-1)^nx^{2n}/(2n)!$ .

Problem 21: Find T_3 for e^x at 0.

Answer: $1+x+x^2/2+x^3/6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use derivatives of e^x.
The verified result is $1+x+x^2/2+x^3/6$ .

Problem 22: Bound alternating-series error after four terms of $\sum(-1)^{n-1}/n$ .

Answer: $\le1/5$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the first omitted term.
The verified result is $\le1/5$ .

Problem 23: For x=t²,y=t³ find dy/dx.

Answer: $3t/2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide dy/dt by dx/dt.
The verified result is $3t/2$ .

Problem 24: Convert r=2cosθ to Cartesian form.

Answer: $(x-1)^2+y^2=1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use r²=2r cosθ.
The verified result is $(x-1)^2+y^2=1$ .

Problem 25: Why must power-series endpoints be tested separately?

Answer: $The ratio/root test is inconclusive when |x-a|=R.$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

At the boundary the limiting ratio equals 1.
The verified result is $The ratio/root test is inconclusive when |x-a|=R.$ .

Common errors

Choosing an integration or convergence test from surface appearance alone.
Skipping endpoint checks.
Claiming a Taylor representation without a convergence or remainder argument.

What is included

Skills assessed

Prerequisites

Exam conditions

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Evaluate ∫xexdx\int xe^x dx∫xexdx.

Problem 2: Evaluate ∫sin⁡3xcos⁡xdx\int\sin^3x\cos xdx∫sin3xcosxdx.

Problem 3: Evaluate ∫1/(x2−1)dx\int1/(x^2-1)dx∫1/(x2−1)dx.

Problem 4: Determine convergence of ∫1∞x−2dx\int_1^\infty x^{-2}dx∫1∞​x−2dx.

Problem 5: Find area between y=x and y=x² on [0,1].

Problem 6: Rotate y=x on [0,2] about x-axis. Find volume.

Problem 7: A force F(x)=4x acts from x=0 to 3. Find work.

Problem 8: Find lim⁡n/(n+1)\lim n/(n+1)limn/(n+1).

Problem 9: Sum ∑n=0∞3(1/2)n\sum_{n=0}^\infty3(1/2)^n∑n=0∞​3(1/2)n.

Problem 10: Test ∑n/(n+1)\sum n/(n+1)∑n/(n+1) for convergence.

Problem 11: Test ∑1/n3/2\sum1/n^{3/2}∑1/n3/2.

Problem 12: Test ∑1/(n2+4)\sum1/(n^2+4)∑1/(n2+4).

Problem 13: Test ∑(3n+1)/(n2+2)\sum(3n+1)/(n^2+2)∑(3n+1)/(n2+2).

Problem 14: Test ∑n!/4n\sum n!/4^n∑n!/4n.

Problem 15: Test ∑(2n/(3n+1))n\sum(2n/(3n+1))^n∑(2n/(3n+1))n.

Problem 16: Test ∑(−1)n−1/n\sum(-1)^{n-1}/n∑(−1)n−1/n.

Problem 17: Classify ∑(−1)n/n2\sum(-1)^n/n^2∑(−1)n/n2.

Problem 18: Find R for ∑xn/5n\sum x^n/5^n∑xn/5n.

Problem 19: Find interval for ∑(x−1)n/n\sum(x-1)^n/n∑(x−1)n/n.

Problem 20: Write the Maclaurin series for cos x.

Problem 21: Find T_3 for e^x at 0.

Problem 22: Bound alternating-series error after four terms of ∑(−1)n−1/n\sum(-1)^{n-1}/n∑(−1)n−1/n.

Problem 23: For x=t²,y=t³ find dy/dx.

Problem 24: Convert r=2cosθ to Cartesian form.

Problem 25: Why must power-series endpoints be tested separately?

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

What is included

Skills assessed

Prerequisites

Exam conditions

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Evaluate ∫xexdx\int xe^x dx∫xexdx.

Problem 2: Evaluate ∫sin⁡3xcos⁡xdx\int\sin^3x\cos xdx∫sin3xcosxdx.

Problem 3: Evaluate ∫1/(x2−1)dx\int1/(x^2-1)dx∫1/(x2−1)dx.

Problem 4: Determine convergence of ∫1∞x−2dx\int_1^\infty x^{-2}dx∫1∞​x−2dx.

Problem 5: Find area between y=x and y=x² on [0,1].

Problem 6: Rotate y=x on [0,2] about x-axis. Find volume.

Problem 7: A force F(x)=4x acts from x=0 to 3. Find work.

Problem 8: Find lim⁡n/(n+1)\lim n/(n+1)limn/(n+1).

Problem 9: Sum ∑n=0∞3(1/2)n\sum_{n=0}^\infty3(1/2)^n∑n=0∞​3(1/2)n.

Problem 10: Test ∑n/(n+1)\sum n/(n+1)∑n/(n+1) for convergence.

Problem 11: Test ∑1/n3/2\sum1/n^{3/2}∑1/n3/2.

Problem 12: Test ∑1/(n2+4)\sum1/(n^2+4)∑1/(n2+4).

Problem 13: Test ∑(3n+1)/(n2+2)\sum(3n+1)/(n^2+2)∑(3n+1)/(n2+2).

Problem 14: Test ∑n!/4n\sum n!/4^n∑n!/4n.

Problem 15: Test ∑(2n/(3n+1))n\sum(2n/(3n+1))^n∑(2n/(3n+1))n.

Problem 16: Test ∑(−1)n−1/n\sum(-1)^{n-1}/n∑(−1)n−1/n.

Problem 17: Classify ∑(−1)n/n2\sum(-1)^n/n^2∑(−1)n/n2.

Problem 18: Find R for ∑xn/5n\sum x^n/5^n∑xn/5n.

Problem 19: Find interval for ∑(x−1)n/n\sum(x-1)^n/n∑(x−1)n/n.

Problem 20: Write the Maclaurin series for cos x.

Problem 21: Find T_3 for e^x at 0.

Problem 22: Bound alternating-series error after four terms of ∑(−1)n−1/n\sum(-1)^{n-1}/n∑(−1)n−1/n.

Problem 23: For x=t²,y=t³ find dy/dx.

Problem 24: Convert r=2cosθ to Cartesian form.

Problem 25: Why must power-series endpoints be tested separately?

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

Problem 1: Evaluate $\int xe^x dx$ .

Problem 2: Evaluate $\int\sin^3x\cos xdx$ .

Problem 3: Evaluate $\int1/(x^2-1)dx$ .

Problem 4: Determine convergence of $\int_1^\infty x^{-2}dx$ .

Problem 8: Find $\lim n/(n+1)$ .

Problem 9: Sum $\sum_{n=0}^\infty3(1/2)^n$ .

Problem 10: Test $\sum n/(n+1)$ for convergence.

Problem 11: Test $\sum1/n^{3/2}$ .

Problem 12: Test $\sum1/(n^2+4)$ .

Problem 13: Test $\sum(3n+1)/(n^2+2)$ .

Problem 14: Test $\sum n!/4^n$ .

Problem 15: Test $\sum(2n/(3n+1))^n$ .

Problem 16: Test $\sum(-1)^{n-1}/n$ .

Problem 17: Classify $\sum(-1)^n/n^2$ .

Problem 18: Find R for $\sum x^n/5^n$ .

Problem 19: Find interval for $\sum(x-1)^n/n$ .

Problem 22: Bound alternating-series error after four terms of $\sum(-1)^{n-1}/n$ .

Problem 1: Evaluate $\int xe^x dx$ .

Problem 2: Evaluate $\int\sin^3x\cos xdx$ .

Problem 3: Evaluate $\int1/(x^2-1)dx$ .

Problem 4: Determine convergence of $\int_1^\infty x^{-2}dx$ .

Problem 8: Find $\lim n/(n+1)$ .

Problem 9: Sum $\sum_{n=0}^\infty3(1/2)^n$ .

Problem 10: Test $\sum n/(n+1)$ for convergence.

Problem 11: Test $\sum1/n^{3/2}$ .

Problem 12: Test $\sum1/(n^2+4)$ .

Problem 13: Test $\sum(3n+1)/(n^2+2)$ .

Problem 14: Test $\sum n!/4^n$ .

Problem 15: Test $\sum(2n/(3n+1))^n$ .

Problem 16: Test $\sum(-1)^{n-1}/n$ .

Problem 17: Classify $\sum(-1)^n/n^2$ .

Problem 18: Find R for $\sum x^n/5^n$ .

Problem 19: Find interval for $\sum(x-1)^n/n$ .

Problem 22: Bound alternating-series error after four terms of $\sum(-1)^{n-1}/n$ .