Logarithmic Integration and Derivatives

This video tutorial explains how to find the indefinite integral of logarithmic functions. It introduces a formula for integrating log base a of u and provides step-by-step examples, including log base 4 of x, log base 5 of x+7, and log base 3 of 5x+9. The tutorial also covers special cases and techniques for handling non-linear functions, emphasizing the use of integration by parts and change of base formula.

What is the formula for the integral of log base a of u?

• u times log base a of u divided by e, divided by u prime

• u times log base a of u, minus e

• u divided by log base a of u, plus e

• u times log base a of u, multiplied by e

When using the change of base formula, what is log base e of x equivalent to?

• x log e

• e log x

• ln x

• log x

In the integration by parts method, what is the integral of u dv equal to?

• u times v minus the integral of v du

• v times u minus the integral of u dv

• v times u plus the integral of u dv

• u times v plus the integral of v du

What is the derivative of ln x?

• x ln x

• 1 over x

• x

• ln x

For the integral of log base 5 of (x + 7), what is the value of u prime?

• 5

• x + 7

• 1

• x

What is the first step in solving the integral of log base 3 of (5x + 9)?

• Multiply by e

• Use integration by parts

• Differentiate the function

• Apply the change of base formula

In the context of logarithms, what does log a minus log b simplify to?

• log a times b

• log a divided by b

• log a plus b

• log a minus b

How can you simplify log base 7 of x to the fourth power?

• 4 times log base 7 of x

• log base 7 of 4x

• log base 7 of x divided by 4

• log base 7 of x plus 4

What is a perfect square trinomial?

• A trinomial that cannot be factored

• A trinomial that can be factored into two identical binomials

• A trinomial with a constant term

• A trinomial with three different terms

What is the integral of log base 2 of (x squared plus 8x plus 16) after factoring?

• 2 times the integral of log base 2 of (x squared)

• The integral of log base 2 of (x + 8)

• The integral of log base 2 of (x squared)

• 2 times the integral of log base 2 of (x + 4)