Definite Integrals Homework: Evaluate & Feedback

[antinerd]

Homework Statement

Evaluate the definite integrals.

Homework Equations

Integral of (t+1)/(t^2+2t+1) dt from 1 to 4 (a=1, b=4)

and

Integral of (xe^(x^2+1)) dx from 0 to 2 (a=0, b=2)

The Attempt at a Solution

I have done them out, just wondering if this is the best way to do them, and perhaps if I made a mistake, it would be nice to know why:

For the first one, I factored and got:

(t+1) / ((t+1)(t+1))

then i canceled and got

1 / (t+1)

Which then means:

ln |t+1| from 1 to 4 = ln |5| - ln |2|

Is that the answer for that?




Now, for the second one, I did the same thing:

antiderivative of (xe^(x^2+1)) = 1/2 (e^(x^2+1)...

right? Then the answer would be from 0 to 2:

e^5 - e^1 = e^4

Any feedback would be great.

 

[neutrino]

Now, for the second one, I did the same thing:

antiderivative of (xe^(x^2+1)) = 1/2 (e^(x^2+1)...

right?

That's right.

Then the answer would be from 0 to 2:

e^5 - e^1 = e^4

Nope. For one, you forgot the factor of 1/2 in front, and also e^a - e^b is not equal to e^(a-b) [actually, that's true for any base.]

The first one's right, BTW.

 

[antinerd]

Thanks.

So if I did the second correctly, it's:

1/2 e^5 - e/2

Right? And I can leave it like that?

 

[learningphysics]

Thanks.

So if I did the second correctly, it's:

1/2 e^5 - e/2

Right? And I can leave it like that?

Yes, looks fine.

 

[neutrino]

If you mean, $$\frac{1}{2}e^5 - \frac{1}{2}e$$, then that's right, although $$\frac{1}{2}\left(e^5 - e\right)$$ would look nicer. :)