Topic 1: Advanced Integration Techniques
1\( x e^x - e^x + C \)
2\( \frac{x^3 \ln x}{3} - \frac{x^3}{9} + C \)
3\( \arctan x + C \)
4\( \frac{1}{2} \ln \left| \frac{x-1}{x+1} \right| + C \)
5\( -x^2 e^{-x} - 2x e^{-x} - 2 e^{-x} + C \)
6\( \frac{x}{2} - \frac{\sin 2x}{4} + C \)
7\( \ln |x-1| - \ln |x+2| + C \)
8\( x \sin x + \cos x + C \)
9\( \frac{x^4 \ln x}{4} - \frac{x^4}{16} + C \)
10\( \frac{1}{2} \arcsin(x^2) + C \)
11\( \frac{1}{2} \ln |(x-1)^2(x+1)| - \frac{1}{x-1} + C \)
Topic 2: Special Functions and Their Integrals
1\( \approx 0.746 \)
2\( -\cos x + C \)
3\( \frac{1}{2} \)
4\( \frac{\pi}{2} \)
5\( \frac{\sqrt{\pi}}{2} \)
6\( \frac{\pi}{2} \)
Topic 3: Higher-Order Differential Equations and Systems of ODEs
1\( y = \sin x \)
2\( y = C_1 e^{2x} + C_2 e^{-2x} \)
3\( y = e^{-t} + t e^{-t} \)
4\( x = \cos t \), \( y = \sin t \)
5\( y = e^t (C_1 \cos t + C_2 \sin t) \)
6\( y = e^{-x} (C_1 + C_2 x) \)
7\( y = C_1 e^{x} + C_2 e^{-x} + C_3 \)
8\( x = e^t (C_1 \cos t + C_2 \sin t) \), \( y = e^t (C_3 \cos t + C_4 \sin t) \)
Topic 4: Functional Equations Involving Calculus
1\( f(x) = e^x \)
2\( f(x) = kx \) (linear)
3\( f(x) = e^{x^2} \)
4\( f(x) = e^x - 1 \)
5\( f(x) = -\frac{1}{x + 1} \)
6\( f(x) = e^{x^2} - e^x \)
Topic 5: Vector Calculus
1\( \frac{\sqrt{2}}{2} \)
2\( \pi \)
3\( 0 \) (Green’s Theorem holds)
4\( \pi \) (Stokes’ Theorem holds)
5\( \pi \) (Green’s Theorem holds)
Topic 6: Complex Analysis Basics
1\( 2\pi i \)
2\( \pi i \)
3\( \frac{\pi}{2} \)
Topic 7: Advanced Problem Solving
1\( \geq 2 \), equality at \( x = 1 \)
2\( 1 \)
3\( 0.25 \) at \( (0.5, 0.5) \)
4\( 0 \)
5\( \leq \frac{1}{(n+1)(m+1)} \)
6\( f(x) = \frac{1}{x} \) (up to normalization)