Integrating Rational Functions to Natural Logarithms

This video tutorial covers the integration of functions that result in natural logarithms. It begins with basic examples, such as the antiderivative of 1/x, and progresses to more complex problems involving substitution and constants. The tutorial also demonstrates the use of long division in integration and provides a variety of examples to illustrate these techniques.

What is the anti-derivative of 1/x?

• e^x + C

• ln|x| + C

• x^2/2

• 1/x + C

How do you integrate 7/x dx?

• 7x + C

• ln|7x| + C

• 7/x + C

• 7 ln|x| + C

What substitution should be made to integrate 1/(x + 5) dx?

• u = x - 5

• u = ln(x)

• u = x + 5

• u = 1/x

What is the anti-derivative of 3/(x - 4) dx?

• 3x - 4 + C

• 3/x - 4 + C

• ln|3x - 4| + C

• 3 ln|x - 4| + C

How do you handle the integral of 8/(5 - 2x) dx?

• 8 ln|5 - 2x| + C

• 4/(5 - 2x) + C

• 4 ln|5 - 2x| + C

• 8/(5 - 2x) + C

What is the anti-derivative of x/(x^2 - 3) dx?

• ln|x^2 - 3| + C

• 1/2 ln|x^2 - 3| + C

• 2 ln|x^2 - 3| + C

• x ln|x^2 - 3| + C

How do you integrate x^2 - 4/x dx?

• x^2 - 4 ln|x| + C

• x^2/2 - 4 ln|x| + C

• x^2/2 + 4 ln|x| + C

• x^2 - 4/x + C

What is the anti-derivative of ln(x^2)/x dx?

• ln(x^2) + C

• ln^2(x) + C

• 1/3 ln^3(x) + C

• 1/2 ln^2(x) + C

How do you integrate (x^2 + 7x + 12)/(x + 3) dx?

• x^2/2 + 4x + C

• x^2/2 + 4 ln|x + 3| + C

• x^2/2 + 7x + C

• x^2/2 + 4x + C

What is the anti-derivative of (x^2 - 2)/(x + 1) dx?

• x^2/2 - x + ln|x + 1| + C

• x^2/2 - x - ln|x + 1| + C

• x^2/2 - ln|x + 1| + C

• x^2/2 - x + C