1
warmup
Antiderivatives
Calculus
Find $\int x\sqrt{3x} dx$.
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2
core
The definite integral
Calculus
\(f(x) = \sqrt{4x+1}, a=0, b=2.\)
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3
hard
Differential equations
Calculus
\(c_{n} = \frac{2}{L} \int_{0}^{L} (Lx - x^{2}) \cos \frac{(2n-1) \pi x}{2L} dx\)
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4
hard
Functions and graphs
Calculus
\(f(x) = \frac{1}{x}.\)
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5
core
Integrals
Calculus
Investigate $\int_{0}^{1} \frac{1}{\sqrt{x}} dx$.
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6
hard
Multivariable calculus
Calculus
Describe the surface parameterized by \( \mathbf{r}(u, v) = \langle u \cos v, u \sin v, u \rangle \), \( -\infty < u < \infty \), \( 0 \le v < 2\pi \).
parametric-surfacesmultivariable-calculusconecartesian-coordinatesparameter-eliminationpythagorean-identitytrigonometric-functionsquadric-surfacesidentify-surface
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Identifying a Parametric Surface as a Cone →
7
core
Parametric equations and vector functions
Calculus
Find the slope and concavity for the curve whose equation is \(x = 2 + \sec \theta, y = 1 + 2 \tan \theta\) at \(\theta = \frac{\pi}{6}\).
parametric-equationsderivativesslopeconcavitysecond-derivativetrigonometric-functionschain-rulecalculus-iievaluate-derivativefind-concavitysecanttangent
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Slope and Concavity of a Parametric Curve Involving Secant and Tangent →
8
hard
Techniques of integration
Calculus
Find a reduction formula for $\int \sec^n x \, dx$ for $n \ge 2$.
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9
warmup
Trigonometric and inverse trigonometric functions
Calculus
Using the $\Delta$-definition, calculate $\frac{d}{dx}(\sin x)$.
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10
core
Vector spaces
Linear algebra
\[ h = \frac{\|\vec{P_1P_2} \cdot \vec{c}\|}{\|\vec{c}\|} \]
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