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Integral of 8x68 x^{6}

The integral of 8x68 x^{6} is 8x77+C\frac{8 x^{7}}{7} + C.

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Graph of f(x) = 8 * x^6 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.620406080100120f(x)
Graph of f(x) = 8 * x^6 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.1428571.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 8 * x6x^{6}.

    āˆ«8x6ā€‰dx\int 8 x^{6}\,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«8x6ā€‰dx=8x77+C\int 8 x^{6}\,dx = \frac{8 x^{7}}{7} + C
  4. 4

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

    ddx[8x77]=392x649\frac{d}{dx}\left[\frac{8 x^{7}}{7}\right] = \frac{392 x^{6}}{49}
  5. 5

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

    āˆ«018x6ā€‰dx=[8x77]01ā‰ˆ1.1428571\int_{0}^{1} 8 x^{6}\,dx = \Big[\frac{8 x^{7}}{7}\Big]_{0}^{1} \approx 1.1428571
LaTeX source of the formulas on this page
\[ \int 8 x^{6}\,dx = \frac{8 x^{7}}{7} + C \]
\[ \int 8 x^{6}\,dx \]
\[ \frac{d}{dx}\left[\frac{8 x^{7}}{7}\right] = \frac{392 x^{6}}{49} \]
\[ \int_{0}^{1} 8 x^{6}\,dx = \Big[\frac{8 x^{7}}{7}\Big]_{0}^{1} \approx 1.1428571 \]

Definite integrals of the same function

IntervalValue
from 0 to 11.1428571
from 0 to 0.50.0089285714

Questions

What is the integral of 8 * x6x^{6}?

The integral of 8x68 x^{6} is 8x77+C\frac{8 x^{7}}{7} + 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 * x6x^{6} from 0 to 1?

About 1.1428571.

More integrals

    Integral of 8 * x^6: 8 * x^7 / 7 + C (with steps) | WordGPT