Practice (26)

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Compute $$\int_0^{\frac{\pi}{4}}(\cos{x} - 2\sin{x}\sin(2x))dx$$

Let $f_0(x)=(\sqrt{e})^x$ , and recursively define $f_{n+1}(x) = f'_n(x)$ for integers $n\ge 0$. Compute $$\sum_{k=0}^{\infty}f_k(1)$$

Consider the parabola $y=ax^2 + 2019x + 2019$. There exists exactly one circle which is centered on the $x$-axis and is tangent to the parabola at exactly two points. It turns out that one of these tangent points is $(0, 2019)$. Find $a$.

What is the smallest natural number $n$ for which the following limit exists?

$$\lim_{x\to 0}\frac{\sin^nx}{\cos^2x(1-\cos{x})^3}$$

Turn the graph of $y=\frac{1}{x}$ by $45^{\circ}$ counter-clockwise and consider the bowl-like top part of the curve (the part above $y=0$). We let a $2D$ fluid accumulate in this $2D$ bowl until the maximum depth of the fluid is $\frac{2\sqrt{2}}{3}$. What’s the area of the fluid used?

Compute $$\int_0^4\frac{dx}{\sqrt{|x-2|}}$$

Compute $$\lim_{x\to 0}\frac{(1-\cos{x})^2}{x^2-x^2\cos^2{x}}$$

Compute $$\int_{-2}^{0}\frac{x^3 + 4x^2 + 7x -20}{x^2+4x+8}dx+\int_0^2\frac{2x^3-7x^2+9x-10}{x^2+4}dx$$

Compute $$\lim_{n\to\infty}n^2\int_0^{\frac{1}{n}}x^{2018x+1}dx$$

Compute $$\int_0^{\pi}\frac{2x\sin{x}}{3+\cos^2x}dx$$

Given that the value $\ln(2)$ is not the root of any polynomial with rational coefficients. For any nonnegative integer $n$, let $p_n(x)$ be the unique polynomial with integer coefficients such that $$p_n(\ln(2)) =\int_1^2 (ln(x))^n dx$$

Compute the value of the $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

There is a unique positive real number $a$ such that the tangent line to $y = x^2 + 1$ at $x = a$ goes through the origin. Compute $a$.

Moor has $\$1000$, and he is playing a gambling game. He gets to pick a number k between $0$ and $1$ (inclusive). A fair coin is then flipped. If the coin comes up heads, Moor is given $5000k$ additional dollars. Otherwise, Moor loses $1000k$ dollars. Moor’s happiness is equal to the log of the amount of money that he has after this gambling game. Find the value of k that Moor should select to maximize his expected happiness.

The set of points $(x, y)$ in the plane satisfying $x^{2/5} + |y| = 1$ form a curve enclosing a region. Compute the area of this region.

Compute the value of $$\int_0^2\sqrt{\frac{4-x}{x}}-\sqrt{\frac{x}{4-x}}dx$$

Compute $$\lim_{x\to\infty}\left[x-x^2\ln\left(\frac{1+x}{x}\right)\right]$$

For a given $x > 0$, let $a_n$ be the sequence defined by $a_1=x$ for $n = 1$ and $a_n = x^{a_{n−1}}$ for $n\ge 2$. Find the largest $x$ for which the limit $\displaystyle\lim_{n\to\infty} a_n$ converges.