I’m having some problems getting to the correct answer for c and d and was wondering if you could someone would look over my work and give me some pointers.
(c) \int x^2ln(x)dx
f(x)=lnx ; g(x)=\frac{1}{3}x^3
f'(x)=\frac{1}{x} ; g'(x)=x^2
lnx\frac{1}{3}x^3-\int\frac{1}{x}\frac{1}{3}x^3dx
ln(x)\frac{1}{3}x^3-\frac{1}{12}ln(x)x^4+C
(d) \int x^3e^{-x^2}dx
u=-x^2
\int -ue^u\frac{du}{2}
\frac{1}{4}(-u^2e^u)+C
\frac{1}{4}x^2e^{-x^2}+C
Am I on the right track?
On (c), here is the line where the value of your expression changes (where you ‘go wrong’):
All of your parts and their derivatives were correct. Where you went wrong was integrating \int\frac{1}{x}\frac{1}{3}x^3dx.
Take your constant multiple \frac{1}{3} to the outside of the integral (using the linearity rule), and then simplify inside of the integral. Remember that \frac{1}{x} is the same as x^{-1}.\ After you simplify inside the integral, finding the antiderivative should be a piece of cake!
I’m not sure exactly what your mistake on (d) was, since you didn’t show all your work. You could have found a wrong value of du. I’ll show my process up to a point:
Now let’s put the integral you came up with in terms of x and dx. This is a good way to check if you’ve substituted correctly. Your new integral was
In terms of x and dx, that would be
which simplifies to
which further simplifies to
This is a different integral from the one you are trying to solve. However, you are very close.
Remember that you can substitute u into the integral as many times as you like.
Once you correctly substitute u and du into your integral, you can solve it using integration by parts. Remember: when your integral is x^n multiplied by e^x, sin(x), or cos(x), let your f(x) = x^n (in this case your “x” will be a “u” and your n will have a value of 1). Then your g'(x) will be the other part (i.e. the other factor) of your integral.
To answer your final question: yes, you are on the right track. Good luck!
Yes, I agree with this for sure. Essentially, it looks like you’ve integrated both parts separately, which is not legal. You appear to have done the same thing on part (d) as well.
Also, let’s please to use the verb “equals” properly in our mathematical writing. For example, we write
rather than
I won’t just assume that you intend to assert that two things are equal to one another just because they are in the same vicinity on your paper.