MAT 101 Lecture Notes - Antiderivative, Product Rule, Partial Fraction Decomposition
Document Summary
Many operations in mathematics have reverses compare addition and subtraction, multiplication and division, and powers and roots. We now know how to find the derivatives of many functions. The reverse operation, anti-derivative will receive our attention in this chapter. A function (cid:1832) is an anti-derivative (or indefinite integral) of (cid:1858) (cid:1856)(cid:1832)(cid:1856)(cid:1876)=(cid:1858), we call (cid:1832) the anti-derivative (or indefinite integral) of (cid:1858). For example, if (cid:1858)((cid:1876))=(cid:1876), we can find its anti-derivative by realizing that for the function (cid:1832)((cid:1876))=(cid:2869)(cid:2870) (cid:1876)(cid:2870) (cid:1856)(cid:1832)(cid:1856)(cid:1876)=(cid:1856)(cid:1856)(cid:1876)(cid:3436)12(cid:1876)(cid:2870)(cid:3440)=12 . 2(cid:1876)=(cid:1876)=(cid:1858)((cid:1876)). Thus (cid:1832)((cid:1876))=(cid:2869)(cid:2870) (cid:1876)(cid:2870) is an anti-derivative of (cid:1858)((cid:1876)) =(cid:1876). However, if (cid:1855) is a constant: (cid:1856)(cid:1856)(cid:1876)(cid:3436)12(cid:1876)(cid:2870)+(cid:1855)(cid:3440)=12 . 2(cid:1876)=(cid:1876) . Since the derivative of a constant is zero. The family of all anti-derivatives of (cid:1876) is thus (cid:2869)(cid:2870)(cid:1876)(cid:2870)+(cid:1855), where (cid:1855) can be any constant. Note that you should always check an anti-derivative (cid:1832) by differentiating it and seeing that you recover (cid:1858).
