Q: A chemical reaction follows $$\frac{dy}{dt} = ky + b$$ with $$k \u003c 0. $$What can you say about the equilibrium solution?

It is stable and solutions approach the equilibrium as $$t \to \infty.$$

Q: Which of the following is the correct setup for evaluating the improper integral $$\$$int_2$$^\infty \frac{1}{x^p} dx$$ for $$p \u003e 1$$?

$$\lim_{b \to \infty} \$$int_2$$^b \frac{1}{x^p} dx$$

Question 1

Evaluate the following integral using an appropriate substitution: 
$$
\int \frac{2x + 5}{x^2$ + 4x + 13} \, dx
$$

Accepted Answer

$$\ln|x^2$ + 4x + 13| + \frac{5}{3} \arctan\left(\frac{x + 2}{3}\right) + C$$

Question 2

Which integration technique is most appropriate for evaluating the integral 
$$
\int x \ln x \, dx
$$?

Accepted Answer

Integration by parts

Question 3

Evaluate the following integral using integration by parts: 
$$
\int x^2$ e$$^{2x}$$ \, dx
$$

Accepted Answer

$$\frac{1}{2} x^2$ e$$^{2x}$$ - x e$$^{2x}$$ + \frac{1}{2} e$$^{2x}$$ + C$$

Question 4

Which substitution is most appropriate for evaluating the integral 
$$
\int \frac{dx}{x^2$ - 5x + 6}
$$?

Accepted Answer

Partial fractions

Question 5

Evaluate the improper integral or state if it diverges: 
$$
\$$int_1$$^{\infty} \frac{\ln x}{x^2$} \, dx
$$

Accepted Answer

$$1 - 0 = 1$$

Question 6

Which of the following integrals would you solve using trigonometric substitution for $$a^2$$ - $$x^2$$?
$$
\int \sqrt{9 - x^2$} \, dx
$$

Accepted Answer

Let $$x = 3 \sin 	heta$$

Question 7

Evaluate the following integral using partial fractions:
$$
\int \frac{3x^2$ + 8x + 2}{x(x + 1)^2$} \, dx
$$

Accepted Answer

$$2 \ln|x| + \ln|x + 1| - \frac{1}{x + 1} + C$$

Question 8

Which of the following best describes the solution method for the differential equation $$\frac{dy}{dt} = t(1 + y^2$)$$ with $$y(0) = 0$$?

Accepted Answer

It is separable and can be solved by integrating both sides after separating variables.

Question 9

A population $$P(t)$$ satisfies the logistic equation $$\frac{dP}{dt} = 0.2P\left(1 - \frac{P}{500}\right)$$ with $$P(0) = 50. $$What is the carrying capacity?

Accepted Answer

$$500$$

Question 10

A cup of coffee cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k(T - 20)$$, $$k \u003e 0. $$What is the equilibrium temperature and is it stable or unstable?

Accepted Answer

Equilibrium at $$T = 20$$, stable

Question 11

Evaluate the following improper integral or state if it diverges:
$$
\$$int_0$$^1 \frac{dx}{x$$^{2/3}$$}
$$

Accepted Answer

$$3$$

Question 12

Which of the following is the correct first step for integrating $$\int \sin^3$ x \cos^4$ x \, dx$$?

Accepted Answer

Use the identity $$\sin^2$ x = 1 - \cos^2$ x$$ and save one $$\sin x$$ for substitution.

Question 13

Given $$\int \frac{e^x}{(9 + e$$^{2x}$$)^{3/2}$$} dx$$, which substitution allows you to use the table formula $$\int \frac{du}{$$(u^2$$ + $$a^2$$)^{3/2}$$} = \frac{u}{a^2$ \sqrt{u^2$ + a^2$}} + C$$?

Accepted Answer

Let $$u = e^x$$

Question 14

Which of the following integrals converges? Justify using the $$p-$$test or comparison test.
$$
\$$int_1$$^{\infty} \frac{dx}{\sqrt{x^3$ + 1}}
$$

Accepted Answer

It converges because the integrand behaves like $$1/x^{3/2}$$ for large $$x.$$

Question 15

Which of the following is the general solution to the first-order linear equation $$y' = ky + b$$?

Accepted Answer

$$y = Ce$$^{kt}$$ + \frac{b}{k}$$

Question 16

Suppose you are given $$\int 	an^3$ x \sec^4$ x \, dx. $$Which substitution is most efficient?

Accepted Answer

Let $$u = 	an x$$

Question 17

A chemical reaction follows $$\frac{dy}{dt} = ky + b$$ with $$k \u003c 0. $$What can you say about the equilibrium solution?

Accepted Answer

It is stable and solutions approach the equilibrium as $$t 	o \infty.$$

Question 18

Which of the following is the correct setup for evaluating the improper integral $$\$$int_2$$^\infty \frac{1}{x^p} dx$$ for $$p \u003e 1$$?

Accepted Answer

$$\lim_{b 	o \infty} \$$int_2$$^b \frac{1}{x^p} dx$$

Question 19

A medication is administered so that its concentration $$C(t) in $$the bloodstream satisfies $$\frac{dC}{dt} = -k(C - 5)$$, $$k \u003e 0. $$What is the long-term concentration as $$t 	o \infty$$?

Accepted Answer

$$5$$

Step-by-Step Calculus Guidance: Integration Techniques & Differential Equations

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