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Integral of 8x38 x^{3}

The integral of 8x38 x^{3} is 8x44+C\frac{8 x^{4}}{4} + C.

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Graph of f(x) = 8 * x^3 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.6-551015202530f(x)
Graph of f(x) = 8 * x^3 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 2.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 8 * x3x^{3}.

    āˆ«8x3ā€‰dx\int 8 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«8x3ā€‰dx=8x44+C\int 8 x^{3}\,dx = \frac{8 x^{4}}{4} + C
  4. 4

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

    ddx[8x44]=128x316\frac{d}{dx}\left[\frac{8 x^{4}}{4}\right] = \frac{128 x^{3}}{16}
  5. 5

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

    āˆ«018x3ā€‰dx=[8x44]01ā‰ˆ2\int_{0}^{1} 8 x^{3}\,dx = \Big[\frac{8 x^{4}}{4}\Big]_{0}^{1} \approx 2
LaTeX source of the formulas on this page
\[ \int 8 x^{3}\,dx = \frac{8 x^{4}}{4} + C \]
\[ \int 8 x^{3}\,dx \]
\[ \frac{d}{dx}\left[\frac{8 x^{4}}{4}\right] = \frac{128 x^{3}}{16} \]
\[ \int_{0}^{1} 8 x^{3}\,dx = \Big[\frac{8 x^{4}}{4}\Big]_{0}^{1} \approx 2 \]

Definite integrals of the same function

IntervalValue
from 0 to 12
from 0 to 0.50.125

Questions

What is the integral of 8 * x3x^{3}?

The integral of 8x38 x^{3} is 8x44+C\frac{8 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 8 * x3x^{3} from 0 to 1?

About 2.

More integrals

    Integral of 8 * x^3: 8 * x^4 / 4 + C (with steps) | WordGPT