Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
The problem states: If $(x+1)f'(x) = f(x) + \frac{1}{x}$, then $f'(\frac{1}{2}) = ?$
Differential EquationsIntegrationPartial Fraction DecompositionCalculus
2025/4/13
The problem asks us to find the value of $l$ if $\int x^2 \, dx = lx + \frac{x^3}{3} + c$, where $c$...
IntegrationDefinite IntegralsCalculusPower Rule
2025/4/13
We are given a set of questions about functions and continuity. We need to find the answers to these...
ContinuityLimitsTrigonometric FunctionsCalculus
2025/4/13
We need to evaluate the definite integral: $\int_{0}^{+\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1...
Definite IntegralCalculusIntegration TechniquesExponential Functions
2025/4/12
We need to evaluate the definite integral $$ \int_{0}^{+\infty} \frac{e^{ax} - e^{bx}}{(1+e^{ax})(1+...
Definite IntegralIntegration TechniquesSubstitutionCalculus
2025/4/12
We are asked to evaluate the definite integral $$ \int_{0}^{+\infty} \frac{e^{ax} - e^{bx}}{(1+e^{ax...
Definite IntegralIntegrationCalculusExponential Functions
2025/4/12
We are asked to evaluate the integral $$ \int_{0}^{+\infty} \frac{[W(x)]^2}{x^2\sqrt{x}} dx $$ where...
Definite IntegralLambert W functionGamma functionIntegration by substitutionCalculus
2025/4/12
We need to evaluate the definite integral: $\int_{0}^{+\infty} \frac{[W(x)]^2}{x^2\sqrt{x}} dx$, whe...
Definite IntegralLambert W functionGamma FunctionIntegration by SubstitutionDivergence
2025/4/12
We need to evaluate the definite integral from $0$ to $+\infty$ of the function $\frac{[W(x)]^2}{x^2...
Definite IntegralLambert W functionSubstitutionGamma Function
2025/4/12
We are asked to evaluate the definite integral $\int_{0}^{\frac{\pi}{6}} \frac{\sin x \sin(x + \frac...
Definite IntegralTrigonometric IdentitiesSubstitutionIntegration Techniques
2025/4/12