Analysis

Problems related to calculus, limits, series, etc.

Problems in this category

The problem defines a sequence $f$ such that for all $n \in \mathbb{N}$, $2f(n+1) + f(n) = 0$, and $...

Sequences and SeriesGeometric SequencesConvergenceLimits

The problem asks to evaluate the integral $\int 2^x dx$. The given solution is $\frac{2^{x+1}}{x+1} ...

IntegrationExponentialsDerivativesCalculus

We are asked to evaluate the limit: $\lim_{x \to 4} \frac{x-4}{x^2-16}$.

LimitsCalculusAlgebraic ManipulationRational Functions

The problem is to evaluate the following expression: $\frac{sin(\frac{2\pi}{3}) + cos(\frac{\pi}{4})...

TrigonometryTrigonometric FunctionsExpression EvaluationSimplificationRationalization

We need to evaluate the double integral $\iint_S \frac{2}{1+x^2} dA$, where $S$ is the triangular re...

Double IntegralsIntegrationCalculusMultivariable CalculusarctanIntegration by Parts

We need to evaluate the double integral $\int_{1}^{2} \int_{0}^{x^2} \frac{y^2}{x} \, dy \, dx$.

Double IntegralIntegration

We are asked to evaluate the iterated integral $\int_{0}^{2} \int_{-x}^{x} e^{-x^2} dy dx$.

Iterated IntegralsCalculusIntegrationDefinite IntegralsSubstitution

The problem asks to evaluate the limit $\lim_{x \to \infty} (\sqrt{x^2+x} - \sqrt{x^2-1})$.

LimitsCalculusRationalizationSquare Roots

We need to evaluate the definite integral $K = \int_0^1 (x^2 + 1)^7 x \, dx$.

Definite IntegralIntegration by SubstitutionPower Rule of Integration

We are asked to find the limit of the function $\frac{e^x - e^{-x}}{2e^x}$ as $x$ approaches $0$. Th...

LimitsExponential FunctionsCalculus

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