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13 results for “integral”

Formulae (13)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

\int u\,dv = uv - \int v\,du

Integration technique from the product rule.

\int_a^b f(x)\,dx = F(b) - F(a)

Connects differentiation and integration.

L = \int_a^b \sqrt{1 + (f'(x))^2}\,dx

Length of a curve y = f(x) from a to b.

S = 2\pi\int_a^b f(x)\sqrt{1+(f'(x))^2}\,dx

Surface area generated by revolving a curve about the x-axis.

\int_a^b f(x)\,dx = \text{signed area under } f

The net area between a curve and the x-axis from a to b.

f_{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx

The mean value of a continuous function over an interval.

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

\int e^x\,dx = e^x + C

Antiderivative of the exponential function.

\int \frac{1}{x}\,dx = \ln|x| + C

Antiderivative of the reciprocal function.

\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx

Mean value of a continuous function over an interval.

V = \pi\int_a^b [f(x)]^2\,dx

Volume of a solid of revolution about the x-axis.

L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx

Length of a curve between two points.