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Integral of 2x22 x^{2}

The integral of 2x22 x^{2} is 2x33+C\frac{2 x^{3}}{3} + C.

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Graph of f(x) = 2 * x^2 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.612345f(x)
Graph of f(x) = 2 * x^2 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.66666667.

How to work it out

  1. 1

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

    āˆ«2x2ā€‰dx\int 2 x^{2}\,dx
  2. 2

    Method: constant multiple and power rule.

  3. 3

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

    āˆ«2x2ā€‰dx=2x33+C\int 2 x^{2}\,dx = \frac{2 x^{3}}{3} + C
  4. 4

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

    ddx[2x33]=18x29\frac{d}{dx}\left[\frac{2 x^{3}}{3}\right] = \frac{18 x^{2}}{9}
  5. 5

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

    āˆ«012x2ā€‰dx=[2x33]01ā‰ˆ0.66666667\int_{0}^{1} 2 x^{2}\,dx = \Big[\frac{2 x^{3}}{3}\Big]_{0}^{1} \approx 0.66666667
LaTeX source of the formulas on this page
\[ \int 2 x^{2}\,dx = \frac{2 x^{3}}{3} + C \]
\[ \int 2 x^{2}\,dx \]
\[ \frac{d}{dx}\left[\frac{2 x^{3}}{3}\right] = \frac{18 x^{2}}{9} \]
\[ \int_{0}^{1} 2 x^{2}\,dx = \Big[\frac{2 x^{3}}{3}\Big]_{0}^{1} \approx 0.66666667 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.66666667
from 0 to 0.50.083333333

Questions

What is the integral of 2 * x2x^{2}?

The integral of 2x22 x^{2} is 2x33+C\frac{2 x^{3}}{3} + 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 2 * x2x^{2} from 0 to 1?

About 0.66666667.

More integrals

    Integral of 2 * x^2: 2 * x^3 / 3 + C (with steps) | WordGPT