Concept Reference

Review these key formulas and facts before attempting the questions.

01
Integration by Parts
★ Must Memorize
∫ u dv = uv − ∫ v du LIATE rule for choosing u: Logarithms, Inverse trig, Algebraic, Trig, Exponential

Choose u to be the function that becomes simpler when differentiated.

Example

Evaluate ∫ x eˣ dx

→ Let u = x, dv = eˣ dx ⟹ xeˣ − eˣ + C

02
Trigonometric Integrals
★ Must Memorize
∫ sinⁿx cosᵐx dx If n odd: factor sin x, use cos²x = 1−sin²x If m odd: factor cos x, use sin²x = 1−cos²x If both even: use half-angle identities sin²x = (1−cos 2x)/2 cos²x = (1+cos 2x)/2
Example

∫ sin³x cos²x dx

→ = −cos³x/3 + cos⁵x/5 + C

03
Trigonometric Substitution
★ Must Memorize
√(a²−x²) → x = a sinθ √(a²+x²) → x = a tanθ √(x²−a²) → x = a secθ

Always draw a right triangle and back-substitute to express the answer in terms of x.

Example

∫ dx / √(4−x²)

→ arcsin(x/2) + C

04
Partial Fractions
★ Must Memorize
Distinct linear: A/(x−a) + B/(x−b) Repeated linear: A/(x−a) + B/(x−a)² Irreducible quad: (Ax+B)/(x²+bx+c)

Degree of numerator must be less than denominator; do polynomial long division first if not.

Example

∫ 1/[(x−1)(x+2)] dx

→ (1/3)ln|x−1| − (1/3)ln|x+2| + C

05
Improper Integrals
★ Must Memorize
∫₁^∞ x⁻ᵖ dx converges iff p > 1 ∫₀¹ x⁻ᵖ dx converges iff p < 1 Comparison Test: 0 ≤ f ≤ g ∫g converges ⟹ ∫f converges ∫f diverges ⟹ ∫g diverges
Example

∫₁^∞ 1/x² dx

→ = 1 (converges, p = 2 > 1)

06
Sequences & Limits
lim n→∞ (1 + 1/n)ⁿ = e lim n→∞ n^(1/n) = 1 lim n→∞ rⁿ = 0 if |r| < 1 lim n→∞ xⁿ/n! = 0 for all x
Example

Does aₙ = (2n+1)/(3n−2) converge?

→ lim = 2/3 (converges)

07
Series Convergence Tests
★ Must Memorize
Geometric: Σ arⁿ converges iff |r| < 1, sum = a/(1−r) p-series: Σ 1/nᵖ converges iff p > 1 Integral Test: f positive, continuous, decreasing Ratio Test: L = lim |a_{n+1}/aₙ| L<1 converges, L>1 diverges, L=1 inconclusive Root Test: L = lim |aₙ|^(1/n) (same rules) Alternating Series: Σ(−1)ⁿbₙ converges if bₙ decreasing and lim bₙ = 0
Example

Σ (3/4)ⁿ from n=0 to ∞

→ Geometric, |r|=3/4 < 1, sum = 1/(1−3/4) = 4

08
Power Series & Radius of Convergence
Σ cₙ(x−a)ⁿ centered at a R = 1 / lim|cₙ₊₁/cₙ| (ratio test) Interval of convergence: check endpoints separately
Example

Σ xⁿ/n! → R = ∞ (entire real line)

→ This is the Taylor series for eˣ

09
Taylor & Maclaurin Series
★ Must Memorize
eˣ = Σ xⁿ/n! all x sin x = Σ (−1)ⁿx^(2n+1)/(2n+1)! all x cos x = Σ (−1)ⁿx^(2n)/(2n)! all x ln(1+x) = Σ (−1)ⁿ⁺¹xⁿ/n |x| ≤ 1, x≠−1 1/(1−x) = Σ xⁿ |x| < 1
Example

First three nonzero terms of sin x Maclaurin

→ x − x³/6 + x⁵/120

10
Parametric Curves & Arc Length
dy/dx = (dy/dt)/(dx/dt) Arc length: L = ∫ √[(dx/dt)² + (dy/dt)²] dt Area under parametric: A = ∫ y(t) x'(t) dt
Example

x = cos t, y = sin t, 0 ≤ t ≤ 2π

→ Circle of radius 1, arc length = 2π

11
Polar Coordinates
x = r cosθ, y = r sinθ, r² = x²+y² Area: A = (1/2)∫ r² dθ Arc length: L = ∫ √(r² + (dr/dθ)²) dθ
Example

Area enclosed by r = 2cosθ

→ A = π (circle of radius 1)

12
Volumes of Revolution
Disk method: V = π∫ [f(x)]² dx Washer method: V = π∫ ([R(x)]²−[r(x)]²) dx Shell method: V = 2π∫ x f(x) dx
Example

Rotate y = √x, 0 ≤ x ≤ 4 about x-axis (disk)

→ V = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π

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Practice Examination

20 multiple-choice questions at exam level. Select your answer — immediate feedback follows. All explanations are collected in the Answer Key at the end.

Q 01 Integration by Parts

Evaluate the integral

∫ x² eˣ dx

Apply integration by parts twice. Let u = x², dv = eˣ dx, so du = 2x dx, v = eˣ.

∫ x²eˣ dx = x²eˣ − ∫ 2xeˣ dx

For the remaining integral, apply IBP again: u = 2x, dv = eˣ dx:

∫ 2xeˣ dx = 2xeˣ − 2eˣ

Combining: x²eˣ − 2xeˣ + 2eˣ + C = eˣ(x² − 2x + 2) + C.

Both A and D are correct forms — A is expanded, D is factored. The answer is D (the factored form most commonly expected).

Q 02 Trig Integrals

Which identity is most useful when evaluating ∫ cos⁴x dx?

When both powers are even, the half-angle (power-reducing) identity is the key tool:

cos⁴x = (cos²x)² = [(1+cos 2x)/2]² = (1+2cos2x+cos²2x)/4

Apply the same identity again to cos²2x = (1+cos4x)/2 to complete the evaluation. Choice B is correct.

Q 03 Trig Substitution

To evaluate the integral

∫ dx / (x² √(x²−9))

which substitution is most appropriate?

The expression under the radical is √(x²−9) = √(x²−3²), which matches the form √(x²−a²). The standard substitution for this form is x = a secθ, so:

x = 3 secθ → √(x²−9) = √(9sec²θ−9) = 3|tanθ|

Choice C is correct.

Q 04 Partial Fractions

The partial fraction decomposition of

5x + 3 ────────────── (x+1)(x²+1)

has the form:

For the irreducible quadratic factor (x²+1), the numerator of the partial fraction must be a general linear polynomial Bx + C, not just a constant. The linear factor (x+1) gets a constant A. Therefore the correct form is:

A/(x+1) + (Bx+C)/(x²+1)

Choice B is correct.

Q 05 Improper Integrals

Which of the following improper integrals converges?

For ∫₁^∞ x⁻ᵖ dx, the integral converges if and only if p > 1.

A: p = 1 → diverges (harmonic) B: p = 1/2 → diverges (p ≤ 1) C: p = 2 → converges ✓ value = 1 D: p = 1/2 → same as B, diverges

Choice C is the only convergent integral.

Q 06 Sequences

Find the limit of the sequence

aₙ = (n² + 3n) / (2n² − 5)

Divide numerator and denominator by n² (highest power):

lim (n²+3n)/(2n²−5) = lim (1 + 3/n)/(2 − 5/n²)

As n → ∞: 3/n → 0 and 5/n² → 0, giving limit = 1/2. Choice B.

Q 07 Geometric Series

Find the sum of the series

∑ (n=0 to ∞) (−2/3)ⁿ

This is a geometric series with a = 1, r = −2/3. Since |r| = 2/3 < 1, it converges.

Sum = a/(1−r) = 1/(1−(−2/3)) = 1/(1+2/3) = 1/(5/3) = 3/5

Choice A is correct.

Q 08 Ratio Test

Apply the Ratio Test to the series

∑ (n=1 to ∞) n! / nⁿ

What is the value of L = lim |aₙ₊₁/aₙ|?

Compute the ratio:

|a_{n+1}/aₙ| = [(n+1)!/(n+1)^(n+1)] / [n!/nⁿ] = (n+1)·nⁿ / (n+1)^(n+1) = nⁿ / (n+1)ⁿ = [n/(n+1)]ⁿ = [1/(1+1/n)]ⁿ → 1/e

Since L = 1/e ≈ 0.368 < 1, the series converges. Choice C.

Q 09 Alternating Series

The alternating series

∑ (n=1 to ∞) (−1)ⁿ⁺¹ / n²

converges. What is the maximum error if we use the first 3 terms as an approximation?

By the Alternating Series Estimation Theorem, the error of approximating an alternating series by its first N terms is bounded by the absolute value of the (N+1)-th term.

Using first 3 terms (N=3), error ≤ |a₄| = 1/4² = 1/16

Choice B.

Q 10 Power Series — Radius

Find the radius of convergence of the power series

∑ (n=0 to ∞) (3ⁿ / (n+1)) · xⁿ

Apply the Ratio Test to find R. With cₙ = 3ⁿ/(n+1):

|a_{n+1}/aₙ| = |3^(n+1)/(n+2)| · |(n+1)/3ⁿ| · |x| = 3·(n+1)/(n+2) · |x| → 3|x|

For convergence, 3|x| < 1, so |x| < 1/3. Therefore R = 1/3. Choice A.

Q 11 Taylor Series

The Maclaurin series for f(x) = eˣ is known. Use it to find the Maclaurin series for

f(x) = e^(−x²)

Substitute −x² in place of x in the Maclaurin series for eˣ = Σ xⁿ/n!:

e^(−x²) = ∑ (−x²)ⁿ/n! = ∑ (−1)ⁿ x^(2n)/n! (n=0 to ∞)

This is choice A. The series converges for all x.

Q 12 Taylor Polynomial

Find the 3rd-degree Taylor polynomial T₃(x) of f(x) = cos x centered at x = 0.

Recall the Maclaurin series for cos x only has even-degree terms:

cos x = 1 − x²/2! + x⁴/4! − ···

The 3rd-degree Taylor polynomial includes terms through degree 3. Since the degree-1 and degree-3 coefficients are both 0 for cos x, T₃(x) = T₂(x):

T₃(x) = 1 − x²/2

Choice A is correct.

Q 13 Parametric Curves

A curve is given parametrically by

x(t) = t³, y(t) = t² − 1

Find dy/dx at the point where t = 2.

dx/dt = 3t², dy/dt = 2t dy/dx = (dy/dt)/(dx/dt) = 2t/(3t²) = 2/(3t) At t=2: dy/dx = 2/(3·2) = 2/6 = 1/3

Choice A is correct.

Q 14 Arc Length

Set up the arc length integral for the parametric curve x = t², y = t³ on [0, 1].

dx/dt = 2t, dy/dt = 3t² L = ∫₀¹ √[(2t)² + (3t²)²] dt = ∫₀¹ √(4t² + 9t⁴) dt

Choice A is correct.

Q 15 Polar Area

Find the area enclosed by the polar curve

r = 3sin θ, 0 ≤ θ ≤ π

The polar area formula is A = (1/2)∫r² dθ:

A = (1/2)∫₀^π (3sinθ)² dθ = (9/2)∫₀^π sin²θ dθ = (9/2)∫₀^π (1−cos2θ)/2 dθ = (9/4)[θ − sin2θ/2]₀^π = (9/4)(π − 0) = 9π/4

Choice A.

Q 16 Volumes — Disk

The region bounded by y = x², y = 0, x = 0, x = 2 is revolved about the x-axis. Find the volume.

V = π∫₀² [f(x)]² dx = π∫₀² x⁴ dx = π[x⁵/5]₀² = π·32/5 = 32π/5

Choice A.

Q 17 Shell Method

Use the shell method to set up the integral for the volume generated by revolving the region bounded by y = x³, y = 0, x = 2 about the y-axis.

Shell method about y-axis: V = 2π∫ (radius)(height) dx = 2π∫ x · f(x) dx.

V = 2π∫₀² x · x³ dx = 2π∫₀² x⁴ dx

Both A and C are equivalent forms. A is the simplified form after multiplying. The setup is C and simplified result is A. The intended answer showing the shell formula structure is C. (A is also correct after simplification.)

Q 18 p-Series & Comparison

Which of the following series diverges?

The p-series Σ 1/nᵖ converges if and only if p > 1.

A: p = 3 > 1 → converges B: p = 3/2 > 1 → converges C: p = 2/3 < 1 → DIVERGES ✓ D: p = 2 > 1 → converges

Choice C.

Q 19 Absolute vs Conditional Convergence

The alternating series

∑ (n=1 to ∞) (−1)ⁿ / √n

is:

Check absolute convergence: Σ 1/√n = Σ n^(−1/2) is a p-series with p = 1/2 < 1, so it diverges. The series is NOT absolutely convergent.

Check the alternating series test: bₙ = 1/√n is positive, decreasing, and lim bₙ = 0. So the alternating series converges by AST.

Since it converges but not absolutely, it is conditionally convergent. Choice B.

Q 20 L'Hôpital / Taylor

Evaluate the limit using either Taylor series or L'Hôpital's Rule:

sin x − x lim ───────── x→0 x³

Using the Maclaurin series for sin x:

sin x = x − x³/6 + x⁵/120 − ··· sin x − x = −x³/6 + x⁵/120 − ··· (sin x − x)/x³ = −1/6 + x²/120 − ···

As x → 0, the higher-order terms vanish, giving the limit = −1/6. Choice C.

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Answer Key & Full Solutions

Detailed step-by-step explanations for all 20 questions

Answer Key & Solutions