Integral of 1/(sqrt(x+1)+sqrt(x))

TL;DR

Learn how to simplify integrals with radical denominators by multiplying the top and bottom by the conjugate, simplifying the integral and eventually getting the answer in power or radical form.

Transcript

ok let's do some math for fun, and here we have this integral and as you can see, this time we put down the sqrt(x+1)+sqrt(x) in the denominator last time, we put this inside of the ln but anyway: pause the video, try this first ok, how did you guys do this? you may want to try to do any kind of substitution, maybe you can do what we did last time ... Read More

Key Insights

• ✖️ To simplify integrals with radical denominators, multiply the top and bottom by the conjugate.
• ✖️ Multiplying by the conjugate eliminates the radical in the denominator, resulting in a simpler integral.
• ✊ The simplified integral can be solved using the power rule.
• ✊ The answer to the integral can be left in power or radical form.
• 🪡 There is no need for substitution when simplifying integrals with radical denominators.
• ❓ Rationalizing the denominator can be useful when doing integrations.
• ✖️ Understanding algebraic techniques, like multiplying by the conjugate, can simplify complex math problems.

Questions & Answers

Q: What is the first step in simplifying an integral with a radical denominator?

The first step is to multiply the top and bottom by the conjugate of the radical denominator.

Q: Why is multiplying by the conjugate useful in simplifying the integral?

Multiplying by the conjugate eliminates the radical in the denominator and simplifies the integral by turning the denominator into a simple 1.

Q: Can we use substitution to solve the integral with a radical denominator?

No, there is no need for substitution in this case. Multiplying by the conjugate is sufficient to simplify the integral.

Q: What are the possible forms of the answer to the simplified integral?

The answer can be left in power form or radical form, depending on personal preference.

Summary & Key Takeaways

• The video teaches how to simplify integrals with radical denominators by multiplying the top and bottom by the conjugate.

• By doing this, the integral is simplified and the denominator becomes a simple 1.

• The simplified integral can then be solved by applying the power rule, resulting in the answer either in power or radical form.