Evaluate the integrals in Exercises 1–8 using integration by p...

Question

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x² ln(x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral x cubed, a lot of XDX using integration by parts. So this problem asks us to use integration by parts. Let's remember that the integral of UDV is equal to uV minus integral of VDU. All that we have to do is assign u equals ln of x, and therefore dV is the remaining part which is x cubed dx in this case. We now want to find the U to begin with, so what we can do is differentiate U equals L and X. DU divided by dx, which is the derivative of ln of x, is going to be 1 divided by x, right? So now duu itself will be equal to multiplying both sides by dx we get dx divided by x. Well done. And now let's identify V, which is the integral of the V. There's the integral of x cubed dx, and we can use the power rule to get x to the power of 4 divided by 4. No necessity to add a constant of integration here. So now the integral of x cubed ln x dx is going to be equal to u multiplied by v. So ln x multiplied by x to the power of 4 divided by 4. We can write x to the power of 4 divided by 4 ln x. And we are going to subtract the integral of VDU. Starting with V, notice that we have a constant 1/4. We can factor it out and integrate x to the power of 4, followed by d u, which is d x divided by x. Now let's simplify slightly x, the power of 4 divided by x. Gives us x cubed and therefore for the final integral we want to integrate x cubed dx, right? So let's go ahead and do that. That's x to the power of 4 divided by 4 ln x minus 1/4. Multiplied by the integral of x cubed dx, we already know the value of that integral is x to the power of 4 divided by 4. And since we are done with integration, we can add a constant of integration c. Now let's multiply to get our final answer of x to the power of 4. Divided by 4. LN X minus X to the power of 4 divided by 4 multiplied by 4 gives us 16 right plus a constant of integration c and this is our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 1–8 using integration by parts.∫ x² ln(x) dx

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Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² ln(x) dx