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Integral of 5x65 x^{6}

The integral of 5x65 x^{6} is 5x77+C\frac{5 x^{7}}{7} + C.

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Graph of f(x) = 5 * x^6 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.610203040506070f(x)
Graph of f(x) = 5 * x^6 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.71428571.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 5 * x6x^{6}.

    āˆ«5x6ā€‰dx\int 5 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«5x6ā€‰dx=5x77+C\int 5 x^{6}\,dx = \frac{5 x^{7}}{7} + C
  4. 4

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

    ddx[5x77]=245x649\frac{d}{dx}\left[\frac{5 x^{7}}{7}\right] = \frac{245 x^{6}}{49}
  5. 5

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

    āˆ«015x6ā€‰dx=[5x77]01ā‰ˆ0.71428571\int_{0}^{1} 5 x^{6}\,dx = \Big[\frac{5 x^{7}}{7}\Big]_{0}^{1} \approx 0.71428571
LaTeX source of the formulas on this page
\[ \int 5 x^{6}\,dx = \frac{5 x^{7}}{7} + C \]
\[ \int 5 x^{6}\,dx \]
\[ \frac{d}{dx}\left[\frac{5 x^{7}}{7}\right] = \frac{245 x^{6}}{49} \]
\[ \int_{0}^{1} 5 x^{6}\,dx = \Big[\frac{5 x^{7}}{7}\Big]_{0}^{1} \approx 0.71428571 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.71428571
from 0 to 0.50.0055803571

Questions

What is the integral of 5 * x6x^{6}?

The integral of 5x65 x^{6} is 5x77+C\frac{5 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 5 * x6x^{6} from 0 to 1?

About 0.71428571.

More integrals

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