<p>find the integral of 2x^(5) * e^(x^2)</p>
<p>notice that x^2 is the exponent. </p>
<p>i need help!!!</p>
<p>u=x^2
du=2xdx
then integral of u^2<em>e^(u)du
by parts then w=u^2 dw=2udu
dv=e^udu v=e^u
so e^u</em>u^2-integral of 2u<em>e^udu
so integral of 2u</em>e^u du again by parts
w=2u dw=2du
dv=e^udu v=e^u</p>
<p>so finally e^u<em>u^2-2</em>u<em>e^u+integral of e^u</em>2<em>du
=e^u</em>u^2-2<em>u</em>e^u+2<em>e^u
=e^(x^2)</em>x^4-2<em>x^2</em>e^(x^2)+2*e^(x^2)</p>
<p>Someone correct me if this is wrong.</p>
<p>I used the tabular method:</p>
<p>I simplified the integral by substituting u = x^2, u’ = 2x and the simplified expression is the integral of x^4e^u. </p>
<p>I eventually get:</p>
<p>x^4(e^x^2)-4x^3(e^x^2)+12x^2(e^x^2)-24x(e^x^2)+24(e^x^2) + C.</p>
<p>^I think you’re just taking the derivative of x^4 and multiplying it by e^(x^2).
You still have to take the integral of e^(x^2) in the tabular method, which is by parts anyway.</p>
<p>I was taking the integral of e^u, which is just the same every time. I wouldn’t be able to use the tabular method right off the bat because taking the anti-derivative of e^x^2 does not yield anything nice. In fact, it yields an error function. It’s not an elementary integration. You would probably need Differential Galois Theory for this function itself.</p>
<p>SAT question?</p>
<p>The anti-derivative for the expression e^(x^2) is not an elementary function and the integral can not be solved exactly. Your best approach to the problem would be to use Simpson’s rule.</p>
<p>Obviously, the integral of 2x<em>e^(x^2)dx is e^(x^2), so I don’t know why you are bringing e^(x^2) into this.<br>
- If we substitute u=x^2 then du=2xdx,
- then we get integral of x^4</em>e^(u)du
- using u=x^2, we get u^2*e^(u)du
- using by parts or tabular method, we can take the derivative of u^2 and the integral of e^udu
- converting back we get as I showed above</p>
<p>Which of these statements is not agreeable?</p>
<p>Also, in CalDud’s calculation, he did the tabular method for x^4*e^u du
This is not correct because you can’t take the derivative of x^4 with respect to u, so you get 0, not x^4, x^3, x^2, x, and 1. Instead, you will end up with x^4, x^2, and 1.</p>
<p>Also, derivative of e^(x^2)<em>x^4-2</em>x^2<em>e^(x^2)+2</em>e^(x^2)+C
is 4x^3<em>e^(x^2)+2x^5</em>e^(x^2)-4x<em>e^(x^2)-4x^3</em>e^(x^2)+4x<em>e^(x^2)+0
So you get 2</em>x^5*e^(x^2)+0 which is the original question.</p>
<p>thanks guys i did both by parts and tabular. </p>
<p>tabular is a lot faster but they both work. I just forgot that the 2x would be eliminated when u substituted the derivative value in the integral.</p>
<p>By parts is the same thing as tabular where you treat one component as the u always and the other as dv always.</p>