Find the integrals. (show steps)
(integral sign) xe^(4x^2)
I think this how is how its done:
(integral sign) xe^(4x^2)
it's a u du problem
let u=4x^2
so, du=8x dx
now you have an x already so all u need is 8 inside and and 1/8 outside the integral
[1/8] (integral sign) [8]xe^(4x^2) dx
1/8(integral sign) e^u du
1/8 e^(4x^2) + C
DONE
Ah. I should have known. Thanks
To find the integral of the function xe^(4x^2), you correctly recognized that it is a u-substitution problem.
1. Start by letting u = 4x^2
This means du = 8x dx (taking the derivative of u with respect to x)
2. Rewrite the integral in terms of u and du.
The integral becomes: ∫ (1/8)e^u du
3. Integrate e^u with respect to u.
The integral of e^u is simply e^u. So, we get: (1/8)e^u
4. Substitute back u with 4x^2.
Replace the u with 4x^2 in the previous result: (1/8)e^(4x^2)
5. Include the constant of integration (C).
Finally, add the constant of integration to the result from step 4: (1/8)e^(4x^2) + C
Therefore, the integral of xe^(4x^2) is (1/8)e^(4x^2) + C, where C is the constant of integration.