How do you evaluate the integral #int x^2e^x#?

Answer 1

#int x^2 e^x dx = e^x(x^2-2x+2)+C#

#int x^2 e^x dx# #= x^2*e^x - int 2x*e^x# #= x^2*e^x -( 2*x * e^x- int 2*e^x)# #= x^2*e^x-(2*x * e^x - 2*e^x) +C# #= e^x(x^2-2x+2)+C#
Explanation: Integrate by parts : #int x^2 e^x dx# Let #u = x^2#, #v = e^x#. You will get #(d v) = e^x*dx#, #(du)=2x*dx#, .
To reduce the power of #x# in the integral, you may rewrite it in this form #int u * d v #. This new expression is equivalent to the original one. (by replacing #x^2# with u and #e^x *dx# with #(d v)#.)
#int u*d v = u*v - int du * v# #int x^2 e^x dx = x^2*e^x - int 2x*e^x#
Now, let's do the partial integration again with #m = 2*x# this time. In this case #d m = 2 *dx#
#int m*d v = m*v - int dm * v# #int 2x*e^x = 2*x * e^x- int 2*e^x = 2*x * e^x - 2*e^x +C# (include the arbritary constant #C# in the expression)
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Answer 2

To evaluate the integral ∫x^2e^x, we use integration by parts. Let u = x^2 and dv = e^x dx. Then, du = 2x dx and v = e^x.

Using the integration by parts formula ∫u dv = uv - ∫v du, we have:

∫x^2e^x dx = x^2e^x - ∫2xe^x dx.

Now, we use integration by parts again for the second integral. Let u = 2x and dv = e^x dx. Then, du = 2 dx and v = e^x.

Applying the formula again, we get:

∫2xe^x dx = 2xe^x - ∫2e^x dx.

Now, we can evaluate the remaining integral:

∫2e^x dx = 2e^x.

Putting it all together, we have:

∫x^2e^x dx = x^2e^x - (2xe^x - 2e^x) = x^2e^x - 2xe^x + 2e^x + C,

where C is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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