1
core
Antiderivatives
Calculus
Evaluate $\int \left( \sqrt[3]{t} + \frac{1}{\sqrt[3]{t}} \right) dt$.
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2
core
Differential equations
Calculus
Solve the eigenvalue problem
\[ y'' + \lambda y = 0, \quad y(0) = 0, \quad y'(L) = 0. \]
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3
core
Multiple integrals
Calculus
Consider the region bounded by the curves $y = \ln x$ and $y = e^x$ in the interval $[1, 2]$. Decompose the region into smaller regions of Type II.
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4
core
Multivariable calculus
Calculus
Let \(\mathbf{F}(x, y) = \langle 4x^3y^4, 4x^4y^3 \rangle\), and suppose that a particle moves from point \((4, 4)\) to \((1, 1)\)
along any smooth curve. Is the work done by \(\mathbf{F}\) on the particle positive, negative, or zero?
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5
core
Parametric equations and vector functions
Calculus
Find \(\frac{dy}{dx}\) for \(x = \sin(t), y = \cos(t)\).
parametric-equationsderivativestrigonometric-functionschain-rulecalculus-ifind-derivativesinecosinetangent
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Derivative of a Parametric Circle →
6
core
Parametric equations and vector functions
Calculus
Solve \(\frac{dy}{dx} = \frac{1+y^2}{1+x^2}.\)
differential-equationsseparable-differential-equationsintegrationinverse-trigonometric-functionsarctangenttangent-addition-formulatrigonometric-identitiescalculus-icalculus-iisolve-differential-equationseparation-of-variablesgeneral-solution
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Solving a Separable Differential Equation with Inverse Tangents →
7
core
Parametric equations and vector functions
Calculus
Find the equation of the parabola with the given properties:
Focus \( (4, 0) \) and directrix \( x = -4 \).
analytic-geometryparabolafocusdirectrixconic-sectionsdistance-formulavertexstandard-formlocus-of-pointsalgebraprecalculusfind-equation
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Equation of a Parabola from Focus and Directrix →
8
core
Techniques of integration
Calculus
If \(n\) is a positive integer, find \( \int_{0}^{2\pi} x \sin nx dx \).
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9
core
Logarithmic and exponential functions
Calculus
Evaluate \(\int_{1}^{2} \frac{x-1}{x^2-2x+2} dx\).
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10
core
Vector spaces
Linear algebra
\( (4, 0, 0) \)
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