1. Evaluate the integral of {xe^3xdx} with a superscript of 2 and lowerscript of 0,, showing full working.?

Question

1 integrate {xe^3xdx} with a superscript of 2 and lowerscript of 0, outside the brackets,, showing full working.

Answer 1

I'm reading this as

2
∫xe^(3x)dx
0

You can do this integral by parts.

Let u=x and dv = e^(3x)dx
Then du=dx and v = ⅓e^(3x)

∫xe^(3x)dx = uv - ∫vdu
∫xe^(3x)dx = ⅓xe^(3x) - ⅓∫e^(3x)dx
∫xe^(3x)dx = ⅓xe^(3x) - (⅓)²e^(3x)

Evaluate this from 0 to 2

⅓*2*e^(3*2) - (⅓)²e^(3*2) - ⅓*0*e^(3*0) + (⅓)²e^(3*0)
⅔e^(6) - (⅓)²e^(6) - 0 + (⅓)²
5/9 * e^6 + 1/9

2
∫xe^(3x)dx = (5e^6 + 1)/9
0

Answer 2

I = ∫ x.e^3x).dx
Integrate by parts using:-

I = ∫ x.e^3x dx = u v - ∫ v (du/dx) dx

u = x and dv/dx = e^3x

I = x.[ (1/3).e^3x) ] - ∫ (1/3).e^3x dx
I = (1/3)( x ) e^3x - (1/9) e^3x limits 0 to 2
I = (2/3) e^6 - (1/9) e^6 + (1/9)
I = (5/9) e^6 + (1/9)
I = (1/9).(5 e^6 + 1)

Answer 3

You need to find a book of standard integrals. You would then find that

Integral x e^ax dx = (e^ax)(ax-1)/a^2

so yours is (e^3x)(3x-1)/9 so just put in the limits.....