Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to st

Evaluate the indefinite integrals by using the given...

Key Concepts

Indefinite Integration of Power Functions

Once an integral is expressed in terms of a new variable, common standard forms emerge, such as integrating functions of the form u^n. The power rule for integration can then be directly applied to evaluate the integral, adding a constant of integration for indefinite integrals.

u-Substitution

This technique involves replacing a part of the integrand with a new variable, usually denoted as u, to simplify the integral. It is particularly useful when the derivative of the chosen substitution is present elsewhere in the integrand, allowing the integral to be rewritten in a standard form.

Chain Rule

In the context of integration, the chain rule is applied in reverse to account for the derivative of the inner function when performing a substitution. This method ensures that the differential in the new variable correctly matches the transformed integrand.

Transcript

00:01 Okay, so we are gonna integrate the following thing, 3x plus 2 times 3x squared plus 4x raised to the fourth.

00:09 This calls for u sub because, hey, we got something trapped by a power.

00:13 So let's let u equal 3x squared plus 4x.

00:19 Now, we are committed to turning this whole integral into terms of u.

00:25 So one thing we need to do is to get this dx to be a du.

00:31 Now there are multiple ways to do this.

00:33 What i like teaching is, hey, let's just derive this u.

00:37 And this is a common step.

00:38 You'll always take the derivative here, 6x plus 4.

00:44 Okay, well, what if we multiply up by dx and divide over by 6x plus 4? if i multiply up by dx and divide over by 6x plus 4, i get dx equals du over 6x plus 4.

01:04 Now, notice that we can just kind of sub that indirectly for dx.

01:09 We're saying the x is equal to this thing.

01:13 But we would still have this to handle.

01:17 Ideally we get some cancellation coming from this term right here.

01:22 Well, one thing we can do to generate that cancellation is to notice, hey, what if we factor out a two from the denominator? well, la, there comes that 3x plus two.

01:37 Now if we just sub in directly, now notice i did not touch this 3x plus two, i called that thing underlined in blue to be used.

01:48 We got u to the fourth.

01:50 And i'm subbing in for dx with du over 2 times 3x plus 4...

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