Question

evaluate integral:

Answer 1

\[\int\limits_{?}^{?}dx/x(x ^{2}+25)\]

Answer 2

use partial fractions to split into two integrals
\[\frac{A}{x} + \frac{Bx+C}{x ^{2}+25} = \frac{1}{x(x ^{2}+25)}\]
(A+B)x^2 +Cx+25A =1
A+B =0 -->A=-B
C=0
25A=1 -->A=1/25
B=-1/25
\[=\frac{1}{25}\int\limits_{}^{}\frac{1}{x}dx - \frac{1}{25}\int\limits_{}^{}\frac{x}{x ^{2}+25} dx\]
Fro second integral use u substitution:
u = x^2+25
du = 2x dx -->dx=du/2x
\[=\frac{1}{25}\ln x - \frac{1}{50}\int\limits_{}^{}\frac{1}{u}du\]
\[=\frac{1}{25}\ln x - \frac{1}{50}\ln |x ^{2}+25| +C\]