不定积分∫（3^x*5^x/25^x-9^x）dx
问题描述：
不定积分∫（3^x*5^x/25^x-9^x）dx
答
∫3^x×5^x/(25^x-9^x)dx
=∫3^x×5^x/[(5²)^x-(3²)^x]dx
=∫[1/(5^x-3^x) - 1/(5^x+3^x)]×5^xdx
=∫{1/[(5/3)^x-1] - 1/[(5/3)^x+1]}×(5/3)^xdx
=∫{1/[(5/3)^x-1] - 1/[(5/3)^x+1]}×d(5/3)^x/ln(5/3)
=[1/ln(5/3)]×[ln|(5/3)^x-1|-ln|(5/3)^x+1|+C
=[1/ln(5/3)]×ln|[(5/3)^x-1]/[(5/3)^x+1]|+C
=[1/ln(5/3)]×ln|(5^x-3^x)/(5^x+3^x]|+C