Integrating Exponentials with Roots that have Roots? (And other small Q's)

[kape]

Hello, I have a few questions! I need clarification on certain points that were not very clear in my calculus book.

----------
Question 1:

I know that \int e^{ax} dx = \frac{1}{a} e^{ax}

But how do you integrate \int e^{ax^2} dx?


-----------
Question 2:

I know that integrating by parts is \int (something) dx= uv - \int vdu

But what if there is a range?

If it is \int_{a}^{b} (something) dx does it equal \left[ uv \right]_{a}^{b} - \int_{a}^{b} vdu or does it simply equal uv - \int_{a}^{b} vdu?


-----------
Question 3:

How do you integrate \int log_ax dx and \int e^{ln|secx|} dx.

In fact, is e^{ln|secx|} reducable?


-----------
Question 4:

I was taught that arcsinx exist only in the range \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] and \left[ \frac{\pi}{2}, \frac{3\pi}{2} \right] (I think because it fails the horizontal test if it isn't in those ranges)

If so, is it possible to integrate \int_{0}^{\pi} xarcsinx dx? (If it is possible, is it because it isn't simply arcsinx but xarcsinx?)


-----------
Question 5:

I am having a lot of problems integrating fuctions with exponents etc that have complex roots. My elementary calculus is shaky at best and I'm taking Advanced Engineering Mathematics (Kreyzig) - I have to. Can anyone recommend me any links or books that may help me?


------------------
Reply to HallsofIvy:

Thank you for your reply! I have a question about your reply on question 1: In my Adv Eng Maths (Kreyzig) book, one of the questions is how to integrate \int xe^{x^2/2} and the answer is e^{x^2/2} + C but I don't understand how to do it!

 

[HallsofIvy]

Answer to question 1: You don't. That integral,
\int e^{ax^2}dx
is well known not to have an elementary integral. In fact, precisely that integral (with a= -1) is important in Statistics and it's integral is defined to be "Erf(x)", the error function.

If it were
/int xe^{ax^2}dx
then you could make the substitution u= e^{ax^2} and have the xdx already for du= 2xdx.

Question 2, Yes, just plug the limits of integration into the formula.

Question 3 seems to have disappeared.

 

[whkoh]

Notice that \frac{d}{dx}\left(\frac{x^2}{2}\right)=x
So it is actually \int{f'\left(x\right)e^{f\left(x\right)}=e^{f\left(x\right)}+c

 

[kape]

Thank you for answering questions 1 & 2.. I think I understand.

Sorry question 3 was deleted, don't quite know how that happened.

Also, I have one more question: (which is kind of similar to question 3)


----------
Question 6

How do you integrate:

\int \frac{1}{x^a} dx

\int \frac{1}{a^x} dx

\int \frac{1}{a^{bx}} dx

\int \frac{1}{a^{bx^{c}}} dx

Should I have learned this somewhere? I don't see these in the integral tables or rules..

 

[HallsofIvy]

\frac{1}{x^a}= x^{-a}
Use the power rule.

\frac{1}{a^x}= a^{-x}
Make the substitution u= -x and then use
\int a^x dx= \frac{a^x}{ln a}

Same thing:
\frac{1}{a^{bx}}= a^{-bx}
Make the substitution u= -bx.

\int \frac{1}{a^{bx^c}}dx
depends strongly on what c is. There is no general anti-derivative.