Did I get the right anti derivate of this function?

[loadingNOW]

So, I have a question about finding the integral of x^3(1+x^2)^1/2

This is basically what I did. Not sure if the answer is right or not.

I did the following u substitution.

u = 1 + x^2
u - 1 = x^2

du = 2xdx

x^2du/2 = 2xdu * x^2

x^2du/2 = x^3du

(u - 1)du/2*u^1/2

distributed the u

1/2*u^3/2 - u^1/2
u^2/2 = u

1/2 ∫ u = 1/4*u^2

= 1/4*(1+x^2)^2

 

[Mark44]

So, I have a question about finding the integral of x^3(1+x^2)^1/2

This is basically what I did. Not sure if the answer is right or not.

I did the following u substitution.

u = 1 + x^2
u - 1 = x^2

du = 2xdx

x^2du/2 = 2xdu * x^2

x^2du/2 = x^3du

(u - 1)du/2*u^1/2

distributed the u

1/2*u^3/2 - u^1/2
u^2/2 = u

1/2 ∫ u = 1/4*u^2

= 1/4*(1+x^2)^2

No. You can always check that your antiderivative (there's no such term as antiderivate) is correct, by differenting it. When you do this, you should get the original integrand.

 

[Mark44]

My first inclination, because of the square root of the sum of squares, would be a trig substitution. Another approach would be to use integration by parts.

If an ordinary substitution works, then that would be preferred, if you can find a suitable one, although I'm not sure that this the way to go in this problem.

Your work is hard to follow, so I didn't try to pick out where you went wrong. Make it easier to follow be starting with x³√(1 + x²) dx, and make you substitution in a systematic way.

 

[loadingNOW]

I was sure I got this right. Well this is a bummer.

 

[Mark44]

Actually, your substitution will work.

(u - 1)du/2*u^1/2

I would write this as
∫(1/2)(u - 1)u^1/2du
= (1/2)∫(u^3/2 - u^1/2)du
Now, carry out the integration and undo your substitution.

 

[loadingNOW]

Yeah, I made a small algebra mistake. Thought you could subtract the exponents but you can't. Only when you divide I suppose. My algebra roots aren't very strong. What do you recommend I do to strengthen my roots so it doesn't hurt me.

 

[Mark44]

If you still have your algebra textbook, spend a little time regularly brushing up on the areas where you feel you're weak. If you don't still have your textbook, take a look at khanacademy.com. They have many videos on all sorts of areas of mathematics.