Changes

=== A Repeated 1st-Degree Denominator ===

=== A Repeated 1st-Degree Denominator ===

The formula for integrals where a first degree polynomial denominator is raised to a power greater than one is much different than the formula above.

The formula for integrals where a first degree polynomial denominator is raised to a power greater than one is much different than the formula above.

<br />Given:

<br /><math>\int \frac{1}{(ax+b)^k}dx</math>

:<math>\int \frac{1}{(ax+b)^k}dx</math>

<br /><math>u = ax+b</math>

:<math>u = ax+b</math>

<br /><math>= \int \frac{1}{u^k} \frac{du}{a} = \frac{1}{a} \int u^{-k} du = \frac{1}{a} \cdot \frac{u^{k-1}}{-k} + C = {-1 \over kau^{k-1}} + C = {-1 \over {ka (ax+b)^{k-1}}} + C</math>

:<math>= \int \frac{1}{u^k} \frac{du}{a} = \frac{1}{a} \int u^{-k)} du = \frac{1}{a} \cdot \frac{u^{k-1}}{-(k-1)} + C = {-1 \over (k-1)au^{k-1}} + C = {-1 \over {(k-1)a (ax+b)^{k-1}}} + C</math>

<br />Note that the above formula '''only''' works if <math>k>1</math>. It does not matter if <math>k</math> is a fraction or even irrational, just as long as it is '''greater than 1'''.

Note that the above formula '''only''' works if <math>k \neq 1</math>.

<br />This means that integrals such as

<br />This means that integrals such as

<br /><math>\int {5 \over {(3x+17)^{12}}}dx</math>

:<math>\int {5 \over {(3x+17)^{12}}}dx</math>

<br />are now very easy:

are now very easy:

<br /><math>= {-5 \over {(3)(12)(3x+17)^{11}}}+C = {-5 \over {36(3x+17)^{11}}}+C</math>

:<math>= {-5 \over {(3)(11)(3x+17)^{11}}}+C = {-5 \over {33(3x+17)^{11}}}+C</math>

== A 2nd-Degree Denominator ==

== A 2nd-Degree Denominator ==