General Integral rules

Homework Statement

Solve (e^(x^2)) , use this result to solve §5x*e^(x^2)dx

Homework Equations

* is multiply
is derive
1.§e^xdx=e^x+C
2.§e^kxdx=1/k*e^kx+C
3.§a^xdx=1/ina*a^x+C (a is a number)

4.§k*f(x)dx=k*§f(x)dx

The Attempt at a Solution

(I'm alittle confused as to how you 'use' the previous one to solve the next.
The previous example simply states this when solving: [in((x^2)+4)]
[in((x^2)+4)]= (in u)= (1/u)*u= 1/((x^2)+4)*2x=(2x)/((x^2)+4)
Then you're going to use this to solve §(3x)/((x^2)+4)dx
gives: §(3x)/((x^2)+4)dx = 3/2§(2x)/((x^2)+4)dx= 3/2*in((x^2)+4)+C.
It's not written, so I suppose they are using it by simply putting 3/2 on the outside, using rule 4)

So my try went:

(e^(x^2))=(eu)*u(um, using the corerule as it is named in norwegian) = e^(x^2)*2x
, which was correct.
Then you use it..
§5x*e^(x^2)dx=(ADSFSDF PRESSING SHIFT FOR A LONG TIME CAN*T WRITE LOL F:: WINDOWS LOL)

malawi_glenn
Homework Helper
Why not learn LaTeX if you hate pressing shift? :-)

what is:

in

integral?

corerule is called "chain rule" in english (I am swedish)

So you got $\frac{d}{dx}(e^{x^2}) = 2xe^{x^2}$, which is correct!

Now what is your result if you use this forumula:
$$\int F(x) dx = f(x) + C$$

if:

$$\frac{dF}{dx} = f$$

?

HallsofIvy
Homework Helper
Hopefully you know that if F'= f, then $F= \int f dx + C$. Since
$$(e^{x^2})'= 2xe^{x^2}$$, what is
$$\int 2x e^{x^2}dx$$?

and, from that, what is
$$\int x e^{x^2}dx[/itex]? oh no, sorry the answer is: (5/2)*e^(x^2)+C... (at work, brb) Where do I find LaTeX? Yeah, I meant chain rule. in is natural logarithm, e^inp=p. My try went: §5x*e^(x^2)dx= 5/2§2x*e^(x^2)dx (cause here is where you 'use it', right? = 5/2*2*(1/(1+1))x^(1+1)*(e^u)*u (chain rule) = 5/2*x^2*e^(x^2)*(1/(2+1))x^(2+1) = 5/2x^2*e^(x^2)*(1/3)x^3+C. (e^u)*u I'm making a fault here? no chain rule in integral? e^(x^2)=e^(x^2)? , but why isn't chain rule being used? (like in §e^(3x-2)dx=(1/3)e^(3x-2)+C.) it looks like it should, heh. And how do the x^2 dissapear.. Last edited: no, it's rule 2? e^(x^2)= 1/x^2*e^(x^2), it would make the x^2 dissapear anyway.. and would make sense at the least. so.. : testing.. [tex]\int 5x e^{x^2}dx = 5/2 \int 2x e^{x^2}dx = 5/2$$\times2\times \frac{1}{1+1}x^1+1

Last edited:
malawi_glenn
Homework Helper
pace

are you to solve

$$\int 5 x e^{x^2}dx$$

?

Then use the derivative of:
$$e^{x^2}$$

And use the:
$$\int F(x) dx = f(x) + C$$

where:
$$\frac{dF}{dx} = f$$

That's all you have to do

yes, i am. Yeah, I figured.

Like this right?:

$$\int 5 x e^{x^2}dx$$ = 5/2§2xe^(x^2) = 5/2§ (and as in opposite of the above e^(x^2)*2x) = (e^(x^2)) , which gives simply 5/2(e^(x^2))+C.

Last edited:
malawi_glenn