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Integral of xex2x e^{x^{2}}

The integral of xex2x e^{x^{2}} is ex22+C\frac{e^{x^{2}}}{2} + C.

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Graph of f(x) = x * exp(x^2) with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.651015f(x)
Graph of f(x) = x * exp(x^2) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.85914091.

How to work it out

  1. 1

    We want an antiderivative of f(x) = x * exp(x2x^{2}).

    ∫xex2 dx\int x e^{x^{2}}\,dx
  2. 2

    Method: substitution u = x².

  3. 3

    That gives the antiderivative; add the constant C because any constant disappears when you differentiate:

    ∫xex2 dx=ex22+C\int x e^{x^{2}}\,dx = \frac{e^{x^{2}}}{2} + C
  4. 4

    Check by differentiating the answer; you must get the original function back:

    ddx[ex22]=2ex22x4\frac{d}{dx}\left[\frac{e^{x^{2}}}{2}\right] = \frac{2 e^{x^{2}} 2 x}{4}
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:

    ∫01xex2 dx=[ex22]01≈0.85914091\int_{0}^{1} x e^{x^{2}}\,dx = \Big[\frac{e^{x^{2}}}{2}\Big]_{0}^{1} \approx 0.85914091
LaTeX source of the formulas on this page
\[ \int x e^{x^{2}}\,dx = \frac{e^{x^{2}}}{2} + C \]
\[ \int x e^{x^{2}}\,dx \]
\[ \frac{d}{dx}\left[\frac{e^{x^{2}}}{2}\right] = \frac{2 e^{x^{2}} 2 x}{4} \]
\[ \int_{0}^{1} x e^{x^{2}}\,dx = \Big[\frac{e^{x^{2}}}{2}\Big]_{0}^{1} \approx 0.85914091 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.85914091
from 0 to 0.50.14201271

Questions

What is the integral of x * exp(x2x^{2})?

The integral of xex2x e^{x^{2}} is ex22+C\frac{e^{x^{2}}}{2} + 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 * exp(x2x^{2}) from 0 to 1?

About 0.85914091.

More integrals

    Integral of x * exp(x^2): exp(x^2) / 2 + C (with steps) | WordGPT