1
hard
Antiderivatives
Calculus
Find $\int \frac{x^3}{\sqrt{x^2 + 25}} dx$.
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2
hard
The definite integral
Calculus
Compute $D_x^2\left(\int_{x^3}^{1789} \frac{1}{t} dt\right)$.
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3
hard
Differential equations
Calculus
\(d_{n} = \frac{2}{L} \int_{0}^{L} x^{2} \sin \frac{(2n-1) \pi x}{2L} dx\)
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4
warmup
Multiple integrals
Calculus
Find the average value of the function $f(x,y) = xy$ over the triangle with vertices $(0, 0)$, $(1, 0)$ and $(1, 3)$.
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5
core
Multivariable calculus
Calculus
Let \( \vec{r}(t) = \langle \cos(\frac{\pi}{2} t), \sin(\frac{\pi}{2} t) \rangle \). Graph \( \vec{r}(t) \) on \( -1 \le t \le 1 \), and find the displace-
ment of \( \vec{r}(t) \) on this interval.
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6
warmup
Parametric equations and vector functions
Calculus
Find the area inside the cardioid defined by the equation \(r = 1 - \cos \theta\).
calculus-iipolar-coordinatesarea-in-polar-coordinatescardioiddefinite-integraltrigonometric-integralshalf-angle-identityintegrationevaluate-area
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Area of a Cardioid →
7
core
Techniques of integration
Calculus
\(\int \ln (x^2 + 1) dx.\)
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8
hard
Logarithmic and exponential functions
Calculus
Prove $e^x = \lim_{u \to +\infty} \left(1 + \frac{x}{u}\right)^u$.
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9
core
Trigonometric and inverse trigonometric functions
Calculus
$x \sin x$
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10
warmup
Vector spaces
Linear algebra
Calculating Dot Products
a. Find the dot product of \( \mathbf{u} = \langle 3, 5, 2 \rangle \) and \( \mathbf{v} = \langle -1, 3, 0 \rangle \).
b. Find the scalar product of \( \mathbf{p} = 10\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \) and \( \mathbf{q} = -2\mathbf{i} + \mathbf{j} + 6\mathbf{k} \).
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