is this integration by parts??

Question

Integrate: X^2*e^3x dc

I'm assuming this is integration by parts, and I would have to integrate twice, but I can't get the answer. My u = x^2, 2du = x dx, dv = e^3x, v = 1/3e^3x. Is this right?

i meant dx, not dc.. sorry

Answer 1

The key to this question is whether or not "dc" at the end of the expression is a typo or is the actual problem!

If "dc" is correct, then you would consider the entire expression a CONSTANT!

Please confirm if "dc" is "dx" instead! Then you have an entirely different approach to the problem!

Answer 2

∫ X² e^3x dx
let u = X² . . . .du = 2x dx
. . dv = e^3x dx . . . . v = 1/3 e^3x
∫ X² e^3x dx = uv - ∫ v du
= 1/3 X² e^3x - 2/3 ∫ x e^3x dx
. . . . . . let u = 2/3 x. . . . . du = 2/3 dx
. . . . . . . . .dv = e^3x dx . . . . v = 1/3 e^3x
∫ X² e^3x dx = 1/3 X² e^3x +2/9 x e^3x - 2/9 ∫ e^3x dx
∫ X² e^3x dx = 1/3 X² e^3x +2/9 x e^3x - 2/27 e^3x + c

Answer 3

du should be 2x dx

assuming you meant ∫ x² e^(3x) dx,

u = x²
du = 2x dx
dv = e^(3x) dx
v = (1/3)e^(3x)

∫ u dv
= uv - ∫ v du
= x²(1/3)e^(3x) - ∫ (1/3)e^(3x)2x dx

u = 2x
du = 2 dx
dv = (1/3)e^(3x) dx
v = (1/9)e^(3x)

∫ (1/3)e^(3x)2x dx
= 2x(1/9)e^(3x) - ∫ (1/9)e^(3x)2 dx
= 2x(1/9)e^(3x) - (2/27)e^(3x)

so ∫ x² e^(3x) dx = x²(1/3)e^(3x) - 2x(1/9)e^(3x) + (2/27)e^(3x)

There's also a nice shortcut: for any polynomial p(x),

∫ p(x) e^x dx = (p(x) - p'(x) + p''(x) - p'''(x) + ... )e^x (continue taking derivatives of the polynomial till you get 0).

So you can set u = 3x, so then du = 3dx and (1/9)u² = x²

∫ x² e^(3x) dx
= (1/27) ∫ u² e^u du
= (1/27) [ u² - 2u + 2 ]e^u
= (1/27) [ 9x² - 6x + 2 ]e^(3x)

Answer 4

∫x^2*e^3x dx

let u = x^2 : du = 2x dx

dv = e^3x : v = (1/3)e^(3x)

∫x^2 e^3x dx = uv - ∫vdu

=>(1/3)x^2e^3x - 2/3∫x e^3x dx

again integrating by parts

u = x : du = dx

dv = e^3x : v = (1/3)e^3x

∫x^2 e^3x dx =(1/3)x^2e^3x - 2/3[xe^3x - 1/3∫e^3x dx]

=>(1/3)x^2e^3x - 2/3[xe^3x - 1/9e^3x ] + c

=>(1/3)x^2*e^3x - (2/3)[xe^3x + (2/27)e^3x ] + c


[Note: You wrote 2du = xdx, it should be du = 2x dx]