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Integral of 2x12 x^{1}

The integral of 2x12 x^{1} is 2x22+C\frac{2 x^{2}}{2} + C.

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Graph of f(x) = 2 * x^1 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.6-1123f(x)
Graph of f(x) = 2 * x^1 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 2 * x1x^{1}.

    āˆ«2x1ā€‰dx\int 2 x^{1}\,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«2x1ā€‰dx=2x22+C\int 2 x^{1}\,dx = \frac{2 x^{2}}{2} + C
  4. 4

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

    ddx[2x22]=8x4\frac{d}{dx}\left[\frac{2 x^{2}}{2}\right] = \frac{8 x}{4}
  5. 5

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

    āˆ«012x1ā€‰dx=[2x22]01ā‰ˆ1\int_{0}^{1} 2 x^{1}\,dx = \Big[\frac{2 x^{2}}{2}\Big]_{0}^{1} \approx 1
LaTeX source of the formulas on this page
\[ \int 2 x^{1}\,dx = \frac{2 x^{2}}{2} + C \]
\[ \int 2 x^{1}\,dx \]
\[ \frac{d}{dx}\left[\frac{2 x^{2}}{2}\right] = \frac{8 x}{4} \]
\[ \int_{0}^{1} 2 x^{1}\,dx = \Big[\frac{2 x^{2}}{2}\Big]_{0}^{1} \approx 1 \]

Definite integrals of the same function

IntervalValue
from 0 to 11
from 0 to 0.50.25

Questions

What is the integral of 2 * x1x^{1}?

The integral of 2x12 x^{1} is 2x22+C\frac{2 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 2 * x1x^{1} from 0 to 1?

About 1.

More integrals

    Integral of 2 * x^1: 2 * x^2 / 2 + C (with steps) | WordGPT