integral of 2^ln(x)

TL;DR

The video demonstrates a quick and simple method to find the integral of 2 to the ln(x) power.

Transcript

this is just a really nice and quick integral the integral of 2 to the ln x power be sure to pause the video and give this a try first okay as we know the base right here is a 2 and the power is the l and x in this kind of situation it's a good idea to write the base in terms of e and here we have the base is just a 2. so let's look at 2 and i want... Read More

Key Insights

• 🍉 Rewriting logarithmic bases in terms of e can simplify integrals.
• ✊ The power rule can be used to integrate functions involving x raised to a power.
• 🥺 Cancelling out common terms can lead to simpler integrals.
• ❓ Strategic substitutions can make integration more efficient.
• 🥺 Applying correct simplification techniques can lead to concise solutions.
• ✊ The derivative of e raised to the power of a function is the original function times the derivative of the exponent.
• ✊ Integrating functions raised to a power involves applying the power rule in reverse.

Questions & Answers

Q: How can the integral of 2 to the ln(x) power be simplified?

To simplify the integral, it is advantageous to rewrite the base 2 as e raised to the power of ln(2), allowing for cancellation and ease of calculation.

Q: What is the relationship between the integral of 2 to the ln(x) power and x raised to the ln(2) power?

The integral of 2 to the ln(x) power is equivalent to the integral of x raised to the ln(2) power due to the cancellation of e raised to the ln(2) power.

Q: What is the final result of the integral of 2 to the ln(x) power?

The integral simplifies to x raised to the ln(2) power multiplied by 2 raised to ln(x), divided by ln(2) plus 1.

Q: Why is it necessary to rewrite the integral using x raised to the ln(2) power and 2 raised to the ln(x) power?

Rewriting the integral in this form allows for the cancellation of e raised to the ln(2) power, simplifying the expression and making it easier to integrate.

Summary & Key Takeaways

• The video presents a step-by-step approach to solve the integral of 2 to the ln(x) power using a strategic substitution.

• By rewriting the base 2 in terms of e, the integral is simplified to e raised to the power of ln(2) multiplied by ln(x).

• Further simplification is achieved by converting the e raised to ln(2) power to just 2, resulting in the integral of x raised to the ln(2) power.