Find the Indefinite integral using integration by parts: S(x^2)(e^x) dxusing the equation Sudv = uv - Svdu I have  figured out that u=x^2 and dv= e^x dx and du = dx and v=e^x.  Assuming that is...

Find the Indefinite integral using integration by parts: S(x^2)(e^x) dx

using the equation Sudv = uv - Svdu I have  figured out that u=x^2 and dv= e^x dx and du = dx and v=e^x.  Assuming that is right, I have calculated the following:

(x^2)(e^x) - Se^x dx

(x^2)(e^x) - e^x +C

(e^x)((x^2) -1) +C

Asked on by kwin13

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to use the following formula to integrate by parts such that:

`int udv = uv - int vdu`

You have made a right selection considering `u = x^2` , since differentiating it, its degree will minimize, but you did not differentiate it right. You should follow the next formula when differentiate `x^2`  such that:

`(x^n)' = n*x^(n-1)`

Reasoning by analogy yields:

`(x^2)' = 2*x^(2-1) = 2x`

Hence, considering `u = x^2`  and `dv = e^x dx`  yieldS:

`u = x^2 => du = 2xdx`

`dv = e^x dx => v = e^x`

`int x^2 e^x dx = x^2*e^x - int 2x*e^x dx`

`int x^2 e^x dx = x^2*e^x - 2 int x* e^x dx`

You need to use parts againto sol,ve the integral `int x*e^x dx`  such that:

`u = x => du = dx`

`dv = e^xdx => v = e^x`

`int x* e^x dx = x*e^x - int e^x dx`

`int x* e^x dx = x*e^x - e^x + c`

Substituting `x*e^x - e^x`  for `int x* e^x dx`  yields:

`int x^2 e^x dx = x^2*e^x - 2 (x*e^x - e^x) + c`

`int x^2 e^x dx = x^2*e^x - 2e^x*(x - 1) +`  c

Factoring out `e^x`  yields:

`int x^2 e^x dx = e^x*(x^2 - 2x + 2) + c`

Hence, evaluating the given indefinite integral using parts yields `int x^2 e^x dx = e^x*(x^2 - 2x + 2) + c` .

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widowspeak's profile pic

widowspeak | (Level 1) Honors

Posted on

You are correct to take u=x^2 and dv = e^x. However, du = 2x dx (i.e. the derivative of x^2 is 2x not 1).

So you should get

x^2 e^x - S 2x e^x dx

S 2x e^x dx itself needs to be integrated by parts

Now u=2x dv=e^x, meaning du = 2, v = e^x

giving S 2x e^x dx = 2x e^x - 2 S e^x dx = 2x e^x - 2e^x + C

Putting this result into the first equation gives

x^2 e^x - 2x e^x + 2e^x + C

 

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