1
hard
Antiderivatives
Calculus
Find $\int \cos^3 5x \sin^2 5x dx$.
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2
core
Differential equations
Calculus
Find $\mathcal{L}\,(u(t-1)(t^2+1))$.
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3
core
Multiple integrals
Calculus
Evaluate the iterated integral \(\iint_{D} (x^2+y^2)dA\) over the region D in the first quadrant between the
functions y = 2x and y = x^2. Evaluate the iterated integral by integrating first with respect to y and then
integrating first with respect to x.
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4
core
Multivariable calculus
Calculus
Use Stokes extquotesingle{} theorem to calculate surface integral $\oint_S \text{curl } \mathbf{F} \cdot d\mathbf{S}$, where $\mathbf{F} = \langle z, x, y \rangle$ and $S$ is the surface as shown in the following figure. The boundary curve, $C$, is oriented clockwise.
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5
core
Parametric equations and vector functions
Calculus
\( r(2 + \sin \theta) = 4 \)
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6
hard
Parametric equations and vector functions
Calculus
\( x^2 + 4xy - 2y^2 - 6 = 0 \)
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7
warmup
Parametric equations and vector functions
Calculus
$3x^2 + 2y^2 - 12y + 6 = 0$
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8
hard
Parametric equations and vector functions
Calculus
\(Vertices located at (\text{-}2, 0), (\text{-}2, \text{-}4) and focus
located at (\text{-}2, -8)\)
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9
hard
Logarithmic and exponential functions
Calculus
Evaluate $\lim_{x \to 0^+} x^{\sin x}$.
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10
warmup
Vector spaces
Linear algebra
Prove Theorem 4.1: Let $V$ be a vector space over a field $K$.
(i) $k0 = 0$. (ii) $0u = 0$. (iii) If $ku = 0$, then $k = 0$ or $u = 0$. (iv) $(-k)u = k(-u) = -ku$.
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