WordGPT

Integral of 2x32 x^{3}

The integral of 2x32 x^{3} is 2x44+C\frac{2 x^{4}}{4} + C.

Explain it step by step in chatMake a short video of it
Graph of f(x) = 2 * x^3 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.62468f(x)
Graph of f(x) = 2 * x^3 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.5.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 2 * x3x^{3}.

    āˆ«2x3ā€‰dx\int 2 x^{3}\,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«2x3ā€‰dx=2x44+C\int 2 x^{3}\,dx = \frac{2 x^{4}}{4} + C
  4. 4

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

    ddx[2x44]=32x316\frac{d}{dx}\left[\frac{2 x^{4}}{4}\right] = \frac{32 x^{3}}{16}
  5. 5

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

    āˆ«012x3ā€‰dx=[2x44]01ā‰ˆ0.5\int_{0}^{1} 2 x^{3}\,dx = \Big[\frac{2 x^{4}}{4}\Big]_{0}^{1} \approx 0.5
LaTeX source of the formulas on this page
\[ \int 2 x^{3}\,dx = \frac{2 x^{4}}{4} + C \]
\[ \int 2 x^{3}\,dx \]
\[ \frac{d}{dx}\left[\frac{2 x^{4}}{4}\right] = \frac{32 x^{3}}{16} \]
\[ \int_{0}^{1} 2 x^{3}\,dx = \Big[\frac{2 x^{4}}{4}\Big]_{0}^{1} \approx 0.5 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.5
from 0 to 0.50.03125

Questions

What is the integral of 2 * x3x^{3}?

The integral of 2x32 x^{3} is 2x44+C\frac{2 x^{4}}{4} + 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 * x3x^{3} from 0 to 1?

About 0.5.

More integrals

    Integral of 2 * x^3: 2 * x^4 / 4 + C (with steps) | WordGPT