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We review substitution and the use of integral tables.

### (Videos) Calculus: Single Variable

Note: The section on the Gompertz Model (7:08–11:31) relates to ideas that we will study later but have not yet seen. It is recommended that you do your best to understand it now and then come back to it again when we study ODEs.

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### Examples

A Basic Table of Integrals
(Note: To use these tables, $k$ and $a$ must represent constants in the integral you wish to compute and cannot depend on the variable of integration.)

 $\displaystyle \int x^k dx = \frac {x^{k+1}}{k+1} + C$ $(k \neq -1)$ $\displaystyle \int \frac {1}{x} dx = \ln |x| + C$ $\displaystyle \int a^x dx = \frac {a^x}{\ln a} + C$ $(a > 0, a \neq 1)$ $\displaystyle \int e^x dx = e^x + C$ $\displaystyle \int x^k \ln |x| dx = \frac {x^{k+1}}{k+1} \ln |x| - \frac {x^{k+1}}{(k+1)^2} + C$ $(k \neq -1)$ $\displaystyle \int \frac {dx}{x^2 + a^2} = \frac {1}{a} \arctan \frac {x}{a} + C$ $(a \neq 0)$ $\displaystyle \int \frac {dx}{\sqrt {a^2 + x^2}} = \ln \left |x + \sqrt {a^2 + x^2}\right | + C$ $\displaystyle \int \frac {dx}{\sqrt {a^2 - x^2}} = \arcsin \frac {x}{a} + C$ $(a > 0, |x| < a)$ $\displaystyle \int \frac {dx}{\sqrt {x^2 - a^2}} = \ln \left | x + \sqrt {x^2-a^2} \right | + C$ $(|x| > |a|)$ $\displaystyle \int \frac {dx}{x \sqrt {x^2 - a^2}} = \frac {1}{a} \mathrm {arcsec} \frac {x}{a} + C$ $(a > 0, x > a)$ $\displaystyle \int \sin x ~dx = - \cos x + C$ $\displaystyle \int \cos x ~dx = \sin x + C$ $\displaystyle \int \tan x ~dx = \ln |\sec x| + C$ $\displaystyle \int \csc x ~dx = - \ln |\csc x + \cot x| + C$ $\displaystyle \int \sec x ~dx = \ln |\sec x + \tan x| + C$ $\displaystyle \int \cot x ~dx = \ln |\sin x| + C$ $\displaystyle \int \sec ^2 x ~ dx = \tan x + C$ $\displaystyle \int \csc ^2 x ~dx = - \cot x + C$ $\displaystyle \int \sec x \tan x dx = \sec x + C$ $\displaystyle \int \csc x \cot x dx = - \csc x + C$