Use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

$$\int _ { 1 } ^ { e } \frac { d y } { y \sqrt { 1 + ( \ln y ) ^ { 2 } } }$$

Alternating current (AC) circuits use complex numbers to represent impedance in ohms. To find the total impedance in a circuit, you need to find the sum of the impedances from each part of the circuit. A circuit has partial impedances of 4.2 + 3i and 5 - 2.2i. What is the total impedance for the circuit?

Three different organic compounds have the formula $\mathrm { C } _ { 3 } \mathrm { H } _ { 8 } \mathrm { O } .$ Only two of these isomers react with $\mathrm { KMnO } _ { 4 }$ (a strong oxidizing agent). What are the names of the products when these isomers react with excess $\mathrm { KMnO } _ { 4 } ?$

Is it possible for a $2 \times 3$ matrix to equal a $3 \times 2$ matrix? Explain.

Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is written on a piece of paper and all the pieces of paper are placed in a hat. One number is selected at random. Determine the probability that the number selected is even.

Sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. y = 1 - 1/x

Explain the difference between skeletal muscle, smooth muscle, and cardiac muscle.

solve the logarithmic equation algebraically. Approximate the result to three decimal places. ln√x+8 = 3

Solve the given initial-value problem.

$$\mathbf{X}^{\prime}=\left(\begin{array}{rrr}{1} & {-12} & {-14} \\ {1} & {2} & {-3} \\ {1} & {1} & {-2}\end{array}\right) \mathbf{X}, \quad \mathbf{X}(0)=\left(\begin{array}{r}{4} \\ {6} \\ {-7}\end{array}\right)$$

Find T(t) and N(t) at the given point. r(t)=

$$\frac { 1 } { 2 } t ^ { 2 } \mathbf { i } + \frac { 1 } { 3 } t ^ { 3 } \mathbf { j }$$

t=1

Find the distance between the points to the nearest tenth. E(-7,0), F(5,8)

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If A is a square matrix such that

$$A^2 = A,$$

then A must be the zero matrix or the identity matrix.

Meredith wants to go on a trip this summer that costs $375. She paid a deposit of$125 and will save an additional $25 per week to pay for the trip. The equation 125 + 25w = 375 can be used to find the number of weeks Meredith will need to save. Describe the steps needed to solve the equation.

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a five-ton satellite to a height of (a) 100 miles above Earth and (b) 300 miles above Earth.