Evaluate. (Be sure to check by differentiating!) ∫x^3 e^x^4...

Key Concepts

Integration by Substitution

This technique involves replacing a part of the integrand with a new variable to simplify the integral. By letting u equal a function within the integrand whose derivative is also present, the integral can be rewritten in terms of u, making it easier to evaluate.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. In the context of integration, it explains why substituting u = g(x) is effective, as the derivative du includes the factor g'(x), which is present in the original integrand.

Verification by Differentiation

After obtaining an antiderivative, differentiating it can serve as a check to ensure the correctness of the integration. If the derivative of the antiderivative equals the original integrand, the solution is verified.

Transcript

00:01 All right, for this problem, we want to evaluate the integral of x cubed times e to the power of x to the power of 4 d x.

00:10 So, we can tell pretty quickly that this isn't something that we know how to evaluate directly.

00:16 So, first step, see if we can do this using substitution.

00:21 So we can do substitution if we can represent this in the form of f of g of x times g prime of x.

00:30 X.

00:31 Well, looking at what we have above, you can say, well, you know, we want to have some function of x inside of another function of x.

00:41 Well, it looks like we have x to the power of four inside of e to the power of x.

00:46 So we can say g of x equals x the power of four.

00:50 F of x equals e to the power of x.

00:55 And, well, that would imply then that g prime of x is going to equal 3x to the power of, or excuse me, not 3x to the power of 4, 4x to the power of 3.

01:12 So, we can see that, well, this doesn't quite work out, but that just means we can refine our guess a little bit and say that, well, let's say f of x equals e to the power of x over 4.

01:27 Putting all of this together then, we would have that our integral is going to be e to the power of x to the power of 4 over 4 times 4x cubed.

01:41 So we have our f of g of x and we have our g prime of x.

01:49 So that means that yes indeed we have something in the form that we can do substitution or use substitution for.

01:55 So, we can add e to the power of x to the power of 4, then that was times x cubed, the x.

02:05 So what we can do here is say u equals x to the power of 4.

02:12 D u then equals 4x cubed, d x.

02:17 So d x is going to equal du over 4x cubed.

02:24 When we do our substitution then, we'll turn this into e to the power of u times x cubed times du over 4x cubed...

Please give Ace some feedback