Limits & Sequences
Does the sequence an = log(n³) / n1/2 converge or diverge? If it converges, find its limit.
The sequence converges to 0. Rewrite as 3·log(n)/√n, then apply L'Hôpital's rule: the limit becomes 6/√n → 0
Evaluate ∫ sin²(x) dx.
x/2 − sin(2x)/4 + C. Use the identity sin²(x) = (1 − cos(2x))/2 and integrate term by term.
Evaluate ∫ 1/(x² − 1) dx using partial fractions.
(1/2)·ln|(x−1)/(x+1)| + C. Decompose 1/(x²−1) = (1/2)·[1/(x−1) − 1/(x+1)] and integrate.
Evaluate ∫ x eˣ dx.
eˣ·(x − 1) + C. Set u = x, dv = eˣ dx, giving x·eˣ − ∫eˣ dx.
Consider the series Σ 1/(n² + n) from n=1 to ∞. Which convergence test is most appropriate, and does the series converge or diverge?
The series converges. Use comparison test.
Evaluate ∫ 1/√(9 − x²) dx
arcsin(x/3) + C. Substitute x = 3·sin(θ), so √(9−x²) = 3·cos(θ) and the integrand simplifies to 1.
Evaluate ∫ (3x + 1)/(x² − x − 2) dx using partial fractions
(7/3)·ln|x−2| + (2/3)·ln|x+1| + C. Factor x²−x−2 = (x−2)(x+1), then set up partial fractions to find A = 7/3 and B = 2/3.
Evaluate ∫ log(x) + sinh2(x) dx
x·ln(x) − x + C. Set u = ln(x), dv = dx, giving x·ln(x) − ∫1 dx
Determine whether Σ n!/5ⁿ from n=1 to ∞ converges or diverges. Name the test you use and justify your choice.
The series diverges. By the Ratio Test.
Evaluate ∫ 1/(x² − 4)^(3/2) dx for x > 2, using a hyperbolic substitution
−x / (4·√(x²−4)) + C. Substitute x = 2·cosh(t): (x²−4)^(3/2) becomes 8·sinh³(t), the integral reduces to (1/4)·∫csch²(t) dt = −(1/4)·coth(t), and back-substituting gives coth(t) = x/√(x²−4).
Evaluate ∫ (2x² + 3) / ((x − 1)(x² + 4)) dx.
log|x−1| + (1/2)·log(x²+4) + (1/2)·arctan(x/2) + C. Decompose as A/(x−1) + (Bx+C)/(x²+4) to find A = 1, B = 1, C = 1, then integrate each term.
Evaluate ∫ x² cosh(x) dx
(x² + 2)·sinh(x) − 2x·cosh(x) + C. Apply IBP twice: first with u = x², dv = sinh(x) dx; then with u = x, dv = cosh(x) dx; substitute back and simplify
Determine whether Σ 1 + sin2(n)/(n^(3/2) + log n) from n=1 to ∞ converges or diverges. Name the test you use and justify why it applies.
The series converges. By the Limit Comparison Test with 1/n^(3/2) (a convergent p-series, p = 3/2).
Evaluate ∫ √(x² + 9) / x dx
√(x²+9) − 3·log|(3 + √(x²+9)) / x| + C. Substitute x = 3·sinh(t): √(x²+9) = 3·cosh(t) and the integrand splits into ∫csch(t) dt + ∫sinh(t) dt.
Evaluate ∫ e3x cos(2x) dx
e3x·(3·cos(2x) + 2·sin(2x)) / 13 + C. Write cos(2x) = Re(e2ix), integrate e(3+2i)x to get e(3+2i)x/(3+2i), rationalise by multiplying by (3−2i)/13, then take the real part.
Evaluate ∫ ex cos(x) dx
eˣ·(cos(x) + sin(x)) / 2 + C. Apply IBP twice to get I = eˣ·cos(x) + eˣ·sin(x) − I, then solve for I
Determine the limit of x² · cos(30π / x)