Friday, 4 September 2026

Find $\int (\tan \theta + \cot \theta)^2 d\theta$.

Evaluate $\sum_{i=1}^{n} (3i^2 + 4)$.

$y_p = u_1e^x + u_2e^{-x}$ is a solution of (A) on $(0,\infty)$ if $u'_1e^x + u'_2e^{-x} = 0$ and $u'_1e^x - u'_2e^{-x} = f(x)$. Solving these two equations yields $u'_1 = \frac{e^{-x}f}{2}$, $u'_2 = -\frac{e^x f}{2}$. The functions $u_1(x) = \frac{1}{2}\int_0^x e^{-t}f(t)dt$ and $u_2(x) = -\frac{1}{2}\int_0^x e^t f(t)dt$ satisfy these conditions. Therefore,

Application to Probability At Sydney extquotesingle s Restaurant, customers must wait an average of 15 minutes for a table. From the time they are seated until they have finished their meal requires an additional 40 minutes, on average. What is the probability that a customer spends less than an hour and a half at the diner, assuming that waiting for a table and completing the meal are independent events?

Given that $f(x, y) = (x - 1)^2 y + (y + 1)^2 x$ is a potential function for $\mathbf{F} = \langle 2xy - 2y + (y + 1)^2, (x - 1)^2 + 2yx + 2x \rangle$, calculate integral $\int_{C} \mathbf{F} \cdot d \mathbf{r}$, where $C$ is the lower half of the unit circle oriented counterclockwise.

Find the polar equation of the conic section with the given properties: Directrix: \(x = 4\); \(e = \frac{1}{5}\)

Establish a reduction formula for $\int x^n e^{ax} \, dx$ for $n \ge 1$.

Find the absolute extrema of $f(x) = \frac{(\ln x)^2}{x}$ on $[1, e]$.

Find the angle at which the curve $y=\frac{1}{3} \sin 3 x$ crosses the $x$-axis.

If the surface of a parabolic reflector is described by equation \(400z = x^2 + y^2,\) find the focal point of the reflector.