What are the dependent and independent variables in the function $h(x)=-2 x^2+3 x-11 ?$

Find the derivative of y with respect to the given independent variable.

$$y=5^{-\cos 2 t}$$

Antidifferentiate.

$$\int \tan ^2 x d x$$

Determine the following:

$$\int x \sqrt{x} d x$$

Seismic Prospecting Exercise are concerned with determining the thickness d of a layer of soil that lies on top of a rock formation. Geologists send two sound pulses from point A to point D separated by a distance s. The first pulse travels directly from A to D along the surface of the earth. The second pulse travels down to the rock formation, then along its surface, and then back up to D (path A B C D ), as in Figure. The pulse travels with velocity $v_1$ in the soil and $v_2$ in the rock.

(a) Show that the time required for the first pulse to travel from A to D is $t_1=s / v_1$.

(b) Show that the time required for the second pulse is

$$t_2=\frac{2 d}{v_1} \sec \theta+\frac{s-2 d \tan \theta}{v_2}$$

provided that

$$\tan \theta \leq \frac{s}{2 d}$$

(Note: If this inequality is not satisfied, then point B does not lie to the left of C.)

(c) Show that $t_2$ is minimized when $\sin \theta=v_1 / v_2$.

Determine the moment of inertia of the area about the $x$ axis. Solve the problem in two ways, using rectangular differential elements: having a thickness $d x$.

Determine the following:

$$\int e d x$$

Find the derivative of y with respect to the given independent variable.

$$y=\log _3(1+\theta \ln 3)$$

Evaluate: $\int \cos 5 x d x$

In the following exercise, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

$$y=\sin x^x$$

Find the derivative of y with respect to the given independent variable.

$$y=x^3 \log _{10} x$$

Integrate.

$$\int x^2 \sin x d x$$

Use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

$$y=\sin x^x$$

Integrate

$$\int(\ln x)^7(d x / x)$$