Integration of Rational Functions into Logarithms By Substitution & Long Division

TL;DR

Learn how to integrate functions that turn into natural log functions, with step-by-step examples and explanations.

Transcript

in this video we're going to talk about how to integrate functions that turn into natural log functions so let's start with this example what is the anti-derivative of 1 /x DX it turns out that this is equal to the natural log of x plus C the derivative of Ln X is 1x so the anti-derivative of 1 /x is Ln X now we need to put an absolute value around... Read More

Key Insights

• ☺️ The anti-derivative of 1/x is ln(x) with absolute value around x.
• ❓ Integrating functions involving constants involves moving the constant to the front.
• 😒 The use of substitution can simplify integration.
• ✋ Long division may be necessary when the degree of the numerator is higher than that of the denominator to integrate.
• 🍉 The anti-derivative of a fraction with two terms in the numerator and one term in the denominator can be simplified by breaking it into smaller fractions.
• 🧑‍💻 Integration of logarithmic functions can involve natural log.
• 🤬 Absolute value symbols are necessary when dealing with potentially negative numbers.

Questions & Answers

Q: What is the anti-derivative of 1/x?

The anti-derivative of 1/x is ln(x) with absolute value around x to account for negative numbers.

Q: How do you find the anti-derivative of 7/x?

The anti-derivative of 7/x is 7ln(x) with absolute value around x.

Q: How do you integrate 1/(x+5)?

The anti-derivative of 1/(x+5) is ln(x+5) with absolute value around x.

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

The anti-derivative of 3/(x-4) is 3ln(x-4).

Q: How do you find the anti-derivative of 8/(5-2x)?

The anti-derivative of 8/(5-2x) is -4ln(5-2x).

Q: How do you integrate x/x^2-3?

The anti-derivative of x/x^2-3 is (1/2)ln(x^2-3) with absolute value around x^2-3.

Q: What is the anti-derivative of x^2-4/x?

The anti-derivative of x^2-4/x is x^2/2 - 4ln(x) + C.

Q: How do you find the anti-derivative of ln(x^2)/x?

The anti-derivative of ln(x^2)/x is (1/3)ln(x^3) + C.

Summary & Key Takeaways

• The anti-derivative of 1/x is ln(x) with absolute value around x to account for negative numbers.

• The anti-derivative of 7/x is 7ln(x) with absolute value around x.

• The anti-derivative of 1/(x+5) is ln(x+5) with absolute value around x.

• The anti-derivative of 3/(x-4) is 3ln(x-4).

• The anti-derivative of 8/(5-2x) is -4ln(5-2x).

• The anti-derivative of x/x^2-3 is (1/2)ln(x^2-3) with absolute value around x^2-3.

• The anti-derivative of x^2-4/x is x^2/2 - 4ln(x) + C.

• The anti-derivative of ln(x^2)/x is (1/3)ln(x^3) + C.

• The anti-derivative of x^2-3x+2/(x+1) is x + 1 - ln(x+1) + C.