5x4 48x2 108xx2 62 dx Answer 3 ln x ln x2 6 CSolution i

5x4 + 48x2 + 108/x(x2+ 6)2 dx Answer 3 ln |x| + ln |x2 + 6| + C

Solution

integral (108+48 x^2+5 x^4)/(x (6+x^2)^2) dx cancelling common terms in the numerator and denominator: = integral (5 x^2+18)/(x (x^2+6)) dx using partial fractions: = integral ((2 x)/(x^2+6)+3/x) dx = 2 integral x/(x^2+6) dx+3 integral 1/x dx substituting u = x^2+6 and du = 2 x dx: = integral 1/u du+3 integral 1/x dx = integral 1/u du+3 log(x) = log(u)+3 log(x)+constant Substituting back for u = x^2+6: = log(x^2+6)+3 log(x)+constant