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Integral of 7x57 x^{5}

The integral of 7x57 x^{5} is 7x66+C\frac{7 x^{6}}{6} + C.

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Graph of f(x) = 7 * x^5 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.6102030405060f(x)
Graph of f(x) = 7 * x^5 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.1666667.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 7 * x5x^{5}.

    āˆ«7x5ā€‰dx\int 7 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«7x5ā€‰dx=7x66+C\int 7 x^{5}\,dx = \frac{7 x^{6}}{6} + C
  4. 4

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

    ddx[7x66]=252x536\frac{d}{dx}\left[\frac{7 x^{6}}{6}\right] = \frac{252 x^{5}}{36}
  5. 5

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

    āˆ«017x5ā€‰dx=[7x66]01ā‰ˆ1.1666667\int_{0}^{1} 7 x^{5}\,dx = \Big[\frac{7 x^{6}}{6}\Big]_{0}^{1} \approx 1.1666667
LaTeX source of the formulas on this page
\[ \int 7 x^{5}\,dx = \frac{7 x^{6}}{6} + C \]
\[ \int 7 x^{5}\,dx \]
\[ \frac{d}{dx}\left[\frac{7 x^{6}}{6}\right] = \frac{252 x^{5}}{36} \]
\[ \int_{0}^{1} 7 x^{5}\,dx = \Big[\frac{7 x^{6}}{6}\Big]_{0}^{1} \approx 1.1666667 \]

Definite integrals of the same function

IntervalValue
from 0 to 11.1666667
from 0 to 0.50.018229167

Questions

What is the integral of 7 * x5x^{5}?

The integral of 7x57 x^{5} is 7x66+C\frac{7 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 7 * x5x^{5} from 0 to 1?

About 1.1666667.

More integrals

    Integral of 7 * x^5: 7 * x^6 / 6 + C (with steps) | WordGPT