Compute the definite integrals. Use a graphing utility to confirm your answers. ∫1^e ln(x^2) d x

Name: Problem 41, Chapter 3: Calculus Volume 2
Uploaded: 2020-07-07T03:42:18-08:00
Duration: 3 min 19 s
Channel: Yaw Asomani
Description: Compute the definite integrals. Use a graphing utility to confirm your answers. ∫1^e ln(x^2) d x

step 1 So let us have this again. Once again, it's a definite integral. We know that with a loss of log

Compute the definite integrals. Use a graphing utility to...

Key Concepts

Integration Techniques

Integration techniques include methods like substitution and integration by parts, which are used to evaluate integrals that are not straightforward. In the context of logarithmic functions, understanding and applying these techniques allows one to find antiderivatives appropriately and compute definite integrals.

Graphical Verification

Graphical verification involves using a graphing utility or calculator to plot the function and visually confirm the computed area under the curve. This step provides an additional layer of assurance that the evaluated integral is correct by comparing the analytical result with the numerical or graphical visualization.

Definite Integral

A definite integral represents the accumulation of a quantity, such as area under a curve, over an interval. It is computed using the Fundamental Theorem of Calculus, which connects the concepts of differentiation and integration by evaluating an antiderivative at the endpoints of the interval.

Properties of Logarithms

Logarithms have several key properties, such as the power rule, which states that ln(a^b) is equal to b * ln(a). This property is essential for simplifying logarithmic expressions before integrating, as it can transform a complex integrand into a more manageable form.

Transcript

00:01 So let us have this again.

00:03 Once again, it's a definite integral.

00:07 We know that with a loss of logarithm, we can rewrite this one as two.

00:17 Okay.

00:18 I can bring this one out.

00:19 So it becomes two.

00:23 Okay, so i have this one to integrate.

00:26 So i'm going to integrate this one.

00:28 I'm going to use my v8.

00:30 Okay.

00:31 This one is a logarithm function.

00:33 So that is definitely going to take the u because the logarithm here is first in terms of which function to choose as u so we're going to let our u be the logarithm function and then the rest is going to be the v and the rest is just the x okay the rest is just the x here and then the u is just going to be 1 over x and then v is just going to be x okay when you take the integral of this one then the integration by part is uv so uv is this is u, this is v.

01:05 We multiply them, okay.

01:07 Then minus integral of v, the u.

01:09 V is x, and then the u is this, okay? so can you see that this one is going to be one? okay, so it's just going to be one.

01:20 So i'm just going to make it the x, okay? one to the power e.

01:25 Here is one to the power, one in e, as upper and lower limit, lower and upper limit, respectively.

01:34 So all i have to do is just find this integral, okay? so we're going to, this integral right here, okay, is this guy right here...

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