Use integration by parts to evaluate the integral: ∫0^1.5 (s^2 + 6)e^s ds

Name: Use integration by parts to evaluate the integral: ∫0^1.5 (s^2 + 6)e^s ds
Uploaded: 2023-11-25T19:59:52-08:00
Duration: 6 min 36 s
Channel: Mengchun Cai
Description: Use integration by parts to evaluate the integral: ∫0^1.5 (s^2 + 6)e^s ds

Step 1: Identify the functions to be integrated and differentiated. In this case, let u = s^2 +

Use integration by parts to evaluate the integral: ∫0^1.5...

Transcript

00:01 Okay, here we are required to compute this indefinite integral.

00:09 It is a fraction of 4 times x cubed minus 12 x squared minus 14 x plus 4 over x squared minus 3 x minus 4 dx.

00:28 Okay, as it is a function in our integral, it is a fraction, not two polynomials.

00:36 We notice it is called a rational function.

00:42 So i notice the degree of the numerator is greater than the degree of the denominator.

00:52 So we want to first simplify this and we want to write this fraction as some summation.

01:02 Okay, this is the first intuition for us to do things like that.

01:06 Then let's try to do that.

01:10 Okay, write this down.

01:15 Use the black color.

01:19 4 times x cubed minus 12 times x squared minus 14 x plus 4 x squared.

01:43 Okay, we want to begin with our process by cutting the information of this term.

01:52 I mean, compare those two leading terms.

01:57 We know we want to multiply 4x so that we can queue some information about the first term.

02:06 To do things like that, we want to insert.

02:11 Let's see.

02:14 Maybe it's better for us to write.

02:20 We want to insert an active 16x here and plot 16x back.

02:29 What do we want to do things like that? because if we combine the first three terms, we know there is a common factor of the denominator and our numerator.

02:41 I mean, it can be written as 4x plus this term.

02:52 16x minus 14x plus 4 over x squared minus 3x.

03:02 It should be 4 minus 4.

03:07 Okay, which is equal to 4x plus 2x plus 4...

Question

Use integration by parts to evaluate the integral: \( \int_0^{1.5} (s^2 + 6)e^s ds \)

Question

Use integration by parts to evaluate the integral: \( \int_0^{1.5} (s^2 + 6)e^s ds \)

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