Topic 1: Advanced Integration Techniques
Easy Questions 🌱
Compute \( \int x e^x \, dx \) using integration by parts.
Evaluate \( \int x^2 \ln x \, dx \) (use integration by parts).
Find \( \int \frac{1}{x^2 + 1} \, dx \) using trigonometric substitution.
Compute \( \int \frac{x}{x^2 - 1} \, dx \) using partial fractions.
Moderate Questions 🌱
Evaluate \( \int x^2 e^{-x} \, dx \) using integration by parts.
Compute \( \int \sin^2 x \, dx \) using trigonometric substitution.
Find \( \int \frac{1}{(x-1)(x+2)} \, dx \) using partial fractions.
Determine \( \int x \cos x \, dx \) using integration by parts.
Hard Questions 🌟
Evaluate \( \int x^3 \ln x \, dx \) using integration by parts.
Compute \( \int \frac{1}{\sqrt{1 - x^4}} \, dx \) using trigonometric substitution.
Find \( \int \frac{x^2 + 1}{(x-1)^2(x+1)} \, dx \) using partial fractions.
Topic 2: Special Functions and Their Integrals
Easy Questions 🌱
Evaluate \( \int_0^1 e^{-x^2} \, dx \) approximately using the Gaussian integral.
Compute \( \int \sin x \, dx \) using the sine integral function.
Moderate Questions 🌱
Find \( \int_0^\infty e^{-x} \cos x \, dx \) using the Laplace transform.
Evaluate \( \int_0^\pi \sin^2 x \, dx \) using the Beta function.
Hard Questions 🌟
Compute \( \int_0^\infty x^2 e^{-x^2} \, dx \) using the Gamma function.
Evaluate \( \int_0^1 x^{-1/2} (1 - x)^{-1/2} \, dx \) using the Beta function.
Topic 3: Higher-Order Differential Equations and Systems of ODEs
Easy Questions 🌱
Solve \( y'' + y = 0 \) with \( y(0) = 0 \), \( y'(0) = 1 \).
Find the general solution of \( y'' - 4y = 0 \).
Moderate Questions 🌱
Solve \( y'' + 2y' + y = 0 \) with \( y(0) = 1 \), \( y'(0) = 0 \).
Find the solution to the system \( \frac{dx}{dt} = y \), \( \frac{dy}{dt} = -x \) with \( x(0) = 0 \), \( y(0) = 1 \).
Hard Questions 🌟
Solve \( y''' - y' = 0 \) with \( y(0) = 1 \), \( y'(0) = 0 \), \( y''(0) = -1 \).
Find the general solution of the system \( \frac{dx}{dt} = 2x + y \), \( \frac{dy}{dt} = x + 2y \).
Topic 4: Functional Equations Involving Calculus
Easy Questions 🌱
Find \( f(x) \) if \( f'(x) = f(x) \) and \( f(0) = 1 \).
Solve \( f(x + y) = f(x) + f(y) \) with \( f(0) = 0 \).
Moderate Questions 🌱
Determine \( f(x) \) if \( f'(x) = x f(x) \) and \( f(0) = 1 \).
Find \( f(x) \) if \( f(x + y) = f(x) f(y) \) with \( f(0) = 1 \).
Hard Questions 🌟
Solve \( f'(x) + f(x)^2 = 0 \) with \( f(0) = 1 \).
Find \( f(x) \) if \( f(x + y) = f(x) e^y + f(y) e^x \) with \( f(0) = 0 \).
Topic 5: Vector Calculus
Easy Questions 🌱
Evaluate \( \int_C x \, ds \) where \( C \) is the line from \( (0, 0) \) to \( (1, 1) \).
Compute \( \iint_S x \, dS \) where \( S \) is the unit disk in the xy-plane.
Moderate Questions 🌱
Evaluate \( \int_C (y \, dx + x \, dy) \) where \( C \) is the circle \( x^2 + y^2 = 1 \).
Compute \( \iint_S \vec{F} \cdot d\vec{S} \) where \( \vec{F} = \langle 0, 0, z \rangle \) and \( S \) is \( z = 1 - x^2 - y^2 \), \( z \geq 0 \).
Hard Questions 🌟
Verify Green’s Theorem for \( \vec{F} = \langle x^2, xy \rangle \) over the square \( [0, 1] \times [0, 1] \).
Apply Stokes’ Theorem to \( \vec{F} = \langle y, z, x \rangle \) over the hemisphere \( x^2 + y^2 + z^2 = 1 \), \( z \geq 0 \).
Topic 6: Complex Analysis Basics
Easy Questions 🌱
Evaluate \( \int_{|z|=1} \frac{1}{z} \, dz \) using contour integration.
Moderate Questions 🌱
Compute \( \int_{|z|=2} \frac{z}{z^2 + 1} \, dz \) using the residue theorem.
Hard Questions 🌟
Evaluate \( \int_{-\infty}^{\infty} \frac{1}{(x^2 + 1)^2} \, dx \) using contour integration.
Topic 7: Advanced Problem Solving
Easy Questions 🌱
Prove \( x + \frac{1}{x} \geq 2 \) for \( x > 0 \) using AM-GM.
Find \( \lim_{x \to 0} \frac{\sin x}{x} \).
Moderate Questions 🌱
Maximize \( xy \) subject to \( x + y = 1 \), \( x, y > 0 \).
Evaluate \( \lim_{x \to \infty} \frac{\ln x}{x} \).
Hard Questions 🌟
Prove \( \int_0^1 x^n (1 - x)^m \, dx \leq \frac{1}{(n+1)(m+1)} \) for \( n, m \geq 0 \).
Find \( f(x) \) maximizing \( \int_0^1 f(x) \ln x \, dx \) subject to \( \int_0^1 f(x) \, dx = 1 \), \( f(x) \geq 0 \).