Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
We are asked to find the derivative of the function $f(x) = (x+2)^x$ using logarithmic differentiati...
DifferentiationLogarithmic DifferentiationChain RuleProduct RuleDerivatives
2025/4/1
We need to evaluate the integral $\int \frac{e^x}{e^{3x}-8} dx$.
IntegrationCalculusPartial FractionsTrigonometric Substitution
2025/4/1
We are asked to evaluate the integral: $\int \frac{e^{2x}}{e^{3x}-8} dx$
IntegrationCalculusSubstitutionPartial FractionsTrigonometric Substitution
2025/4/1
The problem asks us to find the antiderivative of $x \cos(5x)$ given that $f(x) = x \sin(5x)$ and $f...
IntegrationAntiderivativesTrigonometric FunctionsIntegration by Parts (Implicitly)Calculus
2025/3/31
The problem asks to determine the average value of the function $f(x) = \frac{2}{5}x^4$ over the int...
CalculusDefinite IntegralsAverage Value of a FunctionIntegration
2025/3/31
We are asked to evaluate three integrals: i. $\int 3x^2 + \sqrt{x} \, dx$ ii. $\int_{1}^{2} \frac{1}...
IntegrationDefinite IntegralsIndefinite IntegralsPower RuleSubstitution
2025/3/31
The problem asks us to find $\frac{dy}{dx}$ for three different functions. i. $y = x^2 - 3x + 4$, us...
DifferentiationCalculusDerivativesProduct RuleFirst Principles
2025/3/30
We are asked to find the limit of the expression $\lim_{x\to 0} \frac{x+ \sin 3x}{x+ \sin x}$.
LimitsTrigonometric FunctionsL'Hopital's Rule (Implicitly)
2025/3/29
We are given the function $f(x) = x + 2 + \frac{4}{x-1}$ and asked to study its variations on the in...
CalculusDerivativesFunction AnalysisIncreasing/Decreasing IntervalsCritical PointsLocal Maxima/MinimaAsymptotes
2025/3/27
Given a sequence $(u_n)$ such that $u_0 = 3$ and $u_{n+1} = -\frac{1}{3}u_n + 1$. (a) Calculate $u_2...
SequencesSeriesLimitsGeometric SequencesConvergenceArithmetic Sequences
2025/3/27