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Limits

Limits appear in pairs; a number to the right of the top of the integral a number to the right of the bottom of the integral, such as in the following:.

 \int_a^b f(x)\, dx

Indefinite Integrals

Integration where limits are not involved.

Example – \int x^2\, dx=\frac{1}{3}x^3+c

See integrating polynomials.

Definite Integrals

Integration where limits are involved.

Example – \int^3_0 x^2\, dx=\left[\frac{1}{3}x^3\right]^3_0=\left(\frac{1}{3}(3)^3\right)-(\frac{1}{3}(0)^3)=9-0=9.

You might wonder why the integration constant is left off; this is because it cancels when evaluating definite integrals.

See integrating polynomials.