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Integral of 5x35 x^{3}

The integral of 5x35 x^{3} is 5x44+C\frac{5 x^{4}}{4} + C.

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Graph of f(x) = 5 * x^3 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.65101520f(x)
Graph of f(x) = 5 * x^3 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.25.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 5 * x3x^{3}.

    āˆ«5x3ā€‰dx\int 5 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«5x3ā€‰dx=5x44+C\int 5 x^{3}\,dx = \frac{5 x^{4}}{4} + C
  4. 4

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

    ddx[5x44]=80x316\frac{d}{dx}\left[\frac{5 x^{4}}{4}\right] = \frac{80 x^{3}}{16}
  5. 5

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

    āˆ«015x3ā€‰dx=[5x44]01ā‰ˆ1.25\int_{0}^{1} 5 x^{3}\,dx = \Big[\frac{5 x^{4}}{4}\Big]_{0}^{1} \approx 1.25
LaTeX source of the formulas on this page
\[ \int 5 x^{3}\,dx = \frac{5 x^{4}}{4} + C \]
\[ \int 5 x^{3}\,dx \]
\[ \frac{d}{dx}\left[\frac{5 x^{4}}{4}\right] = \frac{80 x^{3}}{16} \]
\[ \int_{0}^{1} 5 x^{3}\,dx = \Big[\frac{5 x^{4}}{4}\Big]_{0}^{1} \approx 1.25 \]

Definite integrals of the same function

IntervalValue
from 0 to 11.25
from 0 to 0.50.078125

Questions

What is the integral of 5 * x3x^{3}?

The integral of 5x35 x^{3} is 5x44+C\frac{5 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 5 * x3x^{3} from 0 to 1?

About 1.25.

More integrals

    Integral of 5 * x^3: 5 * x^4 / 4 + C (with steps) | WordGPT