How to Integrate 2^ln(x) Step-by-Step

TL;DR

To integrate 2^ln(x), rewrite it as e^(ln(2)·ln(x)), which simplifies to x^(ln(2)). Use the reverse power rule to find that the integral equals (2^ln(x)·x) / (ln(2) + 1) plus a constant of integration, C. This substitution significantly simplifies the process.

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: What is the integral of 2^ln(x)?

It equals the integral of x^ln(2), which by the reverse power rule is x^(ln 2 + 1) divided by (ln 2 + 1), plus C. Written another way, the answer is 2^ln(x)·x divided by (ln 2 + 1), plus C. Because ln(2) is just a constant, the whole problem reduces to integrating x raised to a fixed power.

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

Rewrite the base 2 as e^ln(2), since e and ln cancel to give back 2. Then 2^ln(x) becomes (e^ln(2))^ln(x), a power raised to a power, so you multiply the exponents to get e^(ln(x)·ln(2)). This sets up a cancellation that turns the expression into a simple power of x.

Q: Why rewrite the base 2 as e to the ln(2) power?

Writing 2 as e^ln(2) lets you combine and then split the exponents so that an e^ln(x) factor appears. Since e^ln(x) equals x, that factor cancels cleanly. What remains is x^ln(2), a plain constant power of x that is easy to integrate with the power rule.

Q: Why is the integral of 2^ln(x) the same as the integral of x^ln(2)?

After rewriting 2^ln(x) as e^(ln(x)·ln(2)), you regroup it as (e^ln(x))^ln(2). The e^ln(x) part equals x, so the expression collapses to x^ln(2). That is why integrating 2^ln(x) is identical to integrating x^ln(2).

Q: How do you integrate x^ln(2)?

Treat ln(2) as a constant power and apply the power rule backwards: add 1 to the exponent and divide by the new exponent. This gives x^(ln 2 + 1) divided by (ln 2 + 1), plus C. The denominator ln(2) + 1 comes directly from that added 1.

Q: What are the three steps used to solve this integral?

First, rewrite the base 2 as e^ln(2), so the integrand becomes (e^ln(2))^ln(x). Second, multiply the powers to get e^(ln(x)·ln(2)) and regroup it as (e^ln(x))^ln(2). Third, cancel e^ln(x) down to x, leaving x^ln(2), then integrate with the reverse power rule.

Q: How can the final answer be written back in terms of the original expression?

Since x^ln(2) equals the original 2^ln(x), you can rewrite x^(ln 2 + 1) as x^ln(2)·x and swap x^ln(2) back for 2^ln(x). This gives the answer as 2^ln(x)·x divided by (ln 2 + 1), plus C, matching the form of the starting problem.

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.