what is integration of (integrate(x^2)(e^x)dx)?

Question

plz try 2 show work
i m only 14 so i need details u no
and takin calc
so need 2 show work 2 my calc teacher

Answer 1

We need to do this by integration by parts.

The integral of u dv = uv - integral(v du).

So set up a table with u = x^2, so du = 2x dx AND dv = e^x dx so v = e^x.

Now we can rewrite the original integral as x^2 * e^x - integral(2x * e^x dx)

We can bring out the 2, and then we must do integration by parts again.


So set u = x, so du = dx AND dv = e^x dx so v = e^x

Now we can rewrite what we have as x^2 * e^x - 2[x*e^x - integral(e^x dx)]

We can simplify this expression to the following:

x^2 * e^x - 2x*e^x + 2*integral(e^x dx)

The integral of e^x dx = e^x + C.

So we get x^2*e^x - 2x*e^x + 2*e^x + C.

We can now factor out an e^x to get e^x [x^2 - 2x + 2] + C.

And that is our answer.

Answer 2

I think the easiest way to answer integrals in this form is to set up a square

U V
dU dV

then you get

uv - integrate(VdU)

in this case you'll have to iterate the answer several times

starting with
pick
X^2 as your U and e^X as your dV
it'll look like this
X^2 e^X
2x e^X
then you set up your equation like a 7 overlayed onto the box
like.. :
e^x * x^2 - integrate( e^x * 2x )

keep going until there is nothing left to integrate and you'll get

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

hopefully all that makes sense, if you need clarification, ask

Answer 3

To solve this problem, you must use integration by parts.

Integration by parts is defined by:

---------integral(u*dv) = u*v - integral(v*du)--------

When performing integration by parts, you need to define 'u' and 'dv' of your initial integral {integral(x*e^-x*dx)}

For the components of integration by parts, you need to take the derivative of 'u to get 'du' and the integral of 'dv' to get 'v'. When performing integration by parts on an integral with an 'e^x' term, you usually make that the 'dv' term because the integral has an 'e^x' term also.

Thus, we make u = x^2 and dv = e^x*dx

Therefore, du = 2xdx and v = e^x

Now you have all the parts for the integral:

integral(x^2*e^x) = x^2*e^x - 2*integral(xe^x*dx)

Now you need to solve the 2nd part of this problem by integration by parts again. This time, use u=x and dv=e^x*dx

Thus, du=dx, and v = e^x

Now, doing integration by parts, you get:

integral(xe^x*dx) = xe^x - integral(e^xdx)

This simplifies to xe^x-(e^x+C)

Plugging this the the other integral above, the total integral is:

integral(x^2*e^x) = x^2*e^x - 2[xe^x-(e^x+C)]

This can be simplified by taking out a common e^x term:

= e^x[x^2 - 2x + 2] + 2C

Hope this helps

Answer 4

"Hasn't integration ruined our ability to rely on our own people to achieve and progress towards a better standard of living?" The thing is, integration doesn't, and didn't solve very many of the issues that were at the core of black people's problems. Integration only affected superficial issues. The same socio-economic, religious, political, etc. institutions are still in affect and still doing the same damage that they've always done, but now since people have integrated it just doesn't look that way even though everyone is "closer" together physically.

Answer 5

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RE:
what is integration of (integrate(x^2)(e^x)dx)?
plz try 2 show work
i m only 14 so i need details u no
and takin calc
so need 2 show work 2 my calc teacher

Answer 6

you should use integration by parts two times

first round let
p = x^2, dq = e^x dx,

second round use
u = x, dv = e^xdx,

the final result is e^x*(x^2 - 2x +2)

Good Luck