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Integral of 4x54 x^{5}

The integral of 4x54 x^{5} is 4x66+C\frac{4 x^{6}}{6} + C.

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Graph of f(x) = 4 * x^5 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.6-55101520253035f(x)
Graph of f(x) = 4 * x^5 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.66666667.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 4 * x5x^{5}.

    āˆ«4x5ā€‰dx\int 4 x^{5}\,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«4x5ā€‰dx=4x66+C\int 4 x^{5}\,dx = \frac{4 x^{6}}{6} + C
  4. 4

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

    ddx[4x66]=144x536\frac{d}{dx}\left[\frac{4 x^{6}}{6}\right] = \frac{144 x^{5}}{36}
  5. 5

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

    āˆ«014x5ā€‰dx=[4x66]01ā‰ˆ0.66666667\int_{0}^{1} 4 x^{5}\,dx = \Big[\frac{4 x^{6}}{6}\Big]_{0}^{1} \approx 0.66666667
LaTeX source of the formulas on this page
\[ \int 4 x^{5}\,dx = \frac{4 x^{6}}{6} + C \]
\[ \int 4 x^{5}\,dx \]
\[ \frac{d}{dx}\left[\frac{4 x^{6}}{6}\right] = \frac{144 x^{5}}{36} \]
\[ \int_{0}^{1} 4 x^{5}\,dx = \Big[\frac{4 x^{6}}{6}\Big]_{0}^{1} \approx 0.66666667 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.66666667
from 0 to 0.50.010416667

Questions

What is the integral of 4 * x5x^{5}?

The integral of 4x54 x^{5} is 4x66+C\frac{4 x^{6}}{6} + 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 4 * x5x^{5} from 0 to 1?

About 0.66666667.

More integrals

    Integral of 4 * x^5: 4 * x^6 / 6 + C (with steps) | WordGPT