Integral of $x^{2} e^{x}$

Q: What is the integral of x^2 * exp(x)?

The integral of x^2 e^x is ≤ft(x^2 - 2 x + 2) e^x + C.

The integral of $x^{2} e^{x}$ is $\left(x^{2} - 2 x + 2\right) e^{x} + C$ .

\int x^{2} e^{x}\,dx = \left(x^{2} - 2 x + 2\right) e^{x} + C

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Graph of f(x) = x^2 * exp(x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.71828183.

How to work it out

1
We want an antiderivative of f(x) = $x^{2}$ * exp(x).
$\int x^{2} e^{x}\,dx$
2
Method: integration by parts twice.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int x^{2} e^{x}\,dx = \left(x^{2} - 2 x + 2\right) e^{x} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\left(x^{2} - 2 x + 2\right) e^{x}\right] = \left(2 x - 2\right) e^{x} + \left(x^{2} - 2 x + 2\right) e^{x}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} x^{2} e^{x}\,dx = \Big[\left(x^{2} - 2 x + 2\right) e^{x}\Big]_{0}^{1} \approx 0.71828183$

LaTeX source of the formulas on this page

\[ \int x^{2} e^{x}\,dx = \left(x^{2} - 2 x + 2\right) e^{x} + C \]
\[ \int x^{2} e^{x}\,dx \]
\[ \frac{d}{dx}\left[\left(x^{2} - 2 x + 2\right) e^{x}\right] = \left(2 x - 2\right) e^{x} + \left(x^{2} - 2 x + 2\right) e^{x} \]
\[ \int_{0}^{1} x^{2} e^{x}\,dx = \Big[\left(x^{2} - 2 x + 2\right) e^{x}\Big]_{0}^{1} \approx 0.71828183 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.71828183
from 0 to 0.5	0.060901588

Questions

What is the integral of

x^{2}

* exp(x)?

The integral of $x^{2} e^{x}$ is $\left(x^{2} - 2 x + 2\right) e^{x} + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of

x^{2}

* exp(x) from 0 to 1?

About 0.71828183.

Integral of x2exx^{2} e^{x}x2ex

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $x^{2} e^{x}$