\(\displaystyle\int_{0}^{\infty} e^{-2 t} d t\) is

The area of one loop of the graph of the polar equation \(r = 2 \sin(3 \theta)\) is given by which of the following?

\(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{e^{3 x} - 1}{\tan x} =\)

Identify the false statement.

Evaluate \(\displaystyle\int_{0}^{6} \sqrt{6 x - x^2} d x\)

\(\int \ln(x) x^4 d x =\)

Find \(\int e^{m x} \cos(n x) d x\)

Find the arclength of the curve \(y = \ln(\sin(x))\) on the interval \([\dfrac{\pi}{4}, \dfrac{\pi}{2}]\).

The area bounded by the lemniscate with polar equation \(r^2 = 2 \cos(2 \theta)\) is equal to

The graph of the polar equation \(r = \dfrac{1}{\sin \theta - 2 \cos \theta}\) is:

The power series \(x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + ... + \dfrac{x^n}{n} + ...\) converges if and only if:

The power series \((x + 1) - \dfrac{(x + 1)^2}{2!} + \dfrac{(x + 1)^3}{3!} - \dfrac{(x + 1)^4}{4!} + ...\) diverges:

The series \(\displaystyle\sum_{n=0}^{\infty} n! (x - 3)^n\) converges if and only if

The interval of convergence of the series obtained through term by term differentiation of the series \((x - 2) - \dfrac{(x - 2)^2}{4} + \dfrac{(x - 2)^3}{9} - \dfrac{(x - 2)^4}{16} + ...\) is:

The coefficient of \(x^4\) in the Maclaurin series for \(f(x) = e^{-\dfrac{x}{2}}\) is:

The Maclaurin polynomial of order 3 for \(f(x) = \sqrt{1 + x}\) is

The Taylor polynomial of order 3 at \(x = 1\) for \(e^x\) is:

The coefficient of \(\left(x - \dfrac{\pi}{4}\right)^3\) in the Taylor series about \(\dfrac{\pi}{4}\) of \(f(x) = \cos x\) is

The radius of convergence of the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{x^n \cdot n^n}{2^n \cdot n!}\) is: