_substitution: definite integral of exponential function  AP Calculus AB  Khan Academy  Summary and Q&A
TL;DR
The video explains how to calculate the definite integral of x^2 * 2^(x^3) from 0 to 1.
Key Insights
 ðŸ˜‘ Reexpressing the base 2 as e simplifies the problem in the context of finding the definite integral.
 ðŸ¥‹ Usubstitution is a useful technique to transform a complicated expression into a simpler form, making it easier to find the antiderivative.
 ðŸ‘» Evaluating the antiderivative at the upper and lower limits allows us to calculate the definite integral directly.
Transcript
Sal: Let's see if we can calculate the definite integral from zero to one of x squared times two to the x to the third power d x. Like always I encourage you to pause this video and see if you can figure this out on your own. I'm assuming you've had a go at it. There's a couple of interesting things here. The first thing, at least that my brain doe... Read More
Questions & Answers
Q: How is the base 2 reexpressed as e in the problem?
The base 2 is reexpressed as e by finding the natural logarithm of 2, which is approximately 0.693. So, 2 can be represented as e^(0.693).
Q: Why is usubstitution used in this problem?
Usubstitution is used to simplify the expression and make it easier to find the antiderivative. By defining u = x^3 * ln(2), we can rewrite the expression in terms of u and its derivative du.
Q: What is the antiderivative of e^u?
The antiderivative of e^u is simply e^u, as mentioned in the video. So, the integral of e^u du is equal to e^u + C, where C is the constant of integration.
Q: How is the definite integral calculated from 0 to 1?
To calculate the definite integral, you need to evaluate the antiderivative at the upper limit (1) and subtract the evaluation at the lower limit (0). This eliminates the constant of integration and gives you the final result.
Summary & Key Takeaways

The video begins by discussing the need to reexpress the base 2 as e in order to simplify the problem.

Using usubstitution, the video transforms the expression into the antiderivative of e^u, where u = x^3 * ln(2).

After finding the antiderivative, the video evaluates it at 1 and subtracts the evaluation at 0 to obtain the final result of 1/natural log(8).