Calculus Question

Test Preparation
1

<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>

2

<p>you can’t take the integral of e^(x^2) without limits.</p>

3

<p><a href=“http://bektemirov.com/calculus/?func=(2x^5)*(e^(x^2))&opt=intformal”>http://bektemirov.com/calculus/?func=(2x^5)*(e^(x^2))&opt=intformal</a></p>

4

<p>“Possible intermediate steps:
integral 2 e^(x^2) x^5 dx
Factor out constants:
= 2 integral e^(x^2) x^5 dx
For the integrand e^(x^2) x^5, substitute u = x^2 and du = 2 x dx:
= integral e^u u^2 du
For the integrand e^u u^2, integrate by parts, integral f dg = f g- integral g df, where
f = u^2, dg = e^u du,
df = 2 u du, g = e^u:
= e^u u^2-2 integral e^u u du
For the integrand e^u u, integrate by parts, integral f dg = f g- integral g df, where
f = u, dg = e^u du,
df = du, g = e^u:
= e^u u^2-2 e^u u+2 integral e^u du
The integral of e^u is e^u:
= e^u u^2-2 e^u u+2 e^u+constant
Substitute back for u = x^2:
= -2 e^(x^2) x^2+2 e^(x^2)+e^(x^2) x^4+constant
Which is equal to:
= e^(x^2) (x^4-2 x^2+2)+constant”</p>

<p>[integrate</a> (2x^5)*(e^(x^2)) - Wolfram|Alpha](<a href=“integrate (2x^5)*(e^(x^2))]integrate - Wolfram|Alpha”>integrate (2x^5)*(e^(x^2)) - Wolfram|Alpha)</p>

5

<p>thanks 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>