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Integral of x\sqrt{x}

The integral of x\sqrt{x} is 2x1.53+C\frac{2 x^{1.5}}{3} + C.

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Graph of f(x) = sqrt(x) with the area from 0 to 4 shaded-2-11234560.511.522.5f(x)
Graph of f(x) = sqrt(x) with the area from 0 to 4 shaded. The shaded area equals the definite integral, about 5.3333333.

How to work it out

  1. 1

    We want an antiderivative of f(x) = sqrt(x).

    āˆ«xā€‰dx\int \sqrt{x}\,dx
  2. 2

    Method: power rule with exponent 1/2.

  3. 3

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

    āˆ«xā€‰dx=2x1.53+C\int \sqrt{x}\,dx = \frac{2 x^{1.5}}{3} + C
  4. 4

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

    ddx[2x1.53]=9x0.59\frac{d}{dx}\left[\frac{2 x^{1.5}}{3}\right] = \frac{9 x^{0.5}}{9}
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 4:

    āˆ«04xā€‰dx=[2x1.53]04ā‰ˆ5.3333333\int_{0}^{4} \sqrt{x}\,dx = \Big[\frac{2 x^{1.5}}{3}\Big]_{0}^{4} \approx 5.3333333
LaTeX source of the formulas on this page
\[ \int \sqrt{x}\,dx = \frac{2 x^{1.5}}{3} + C \]
\[ \int \sqrt{x}\,dx \]
\[ \frac{d}{dx}\left[\frac{2 x^{1.5}}{3}\right] = \frac{9 x^{0.5}}{9} \]
\[ \int_{0}^{4} \sqrt{x}\,dx = \Big[\frac{2 x^{1.5}}{3}\Big]_{0}^{4} \approx 5.3333333 \]

Definite integrals of the same function

IntervalValue
from 0 to 45.3333333
from 0 to 21.8856181

Questions

What is the integral of sqrt(x)?

The integral of x\sqrt{x} is 2x1.53+C\frac{2 x^{1.5}}{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 sqrt(x) from 0 to 4?

About 5.3333333.

More integrals

    Integral of sqrt(x): 2 * x^1.5 / 3 + C (with steps) | WordGPT