不定积分∫（3^x*5^x/25^x-9^x）dx

∫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

∫3^x×5^x/(25^x-9^x)dx
=1／2∫3^x×5^x/[(5²)^x-(3²)^x]dx
=1/2∫[1/(5^x-3^x) - 1/(5^x 3^x)]×5^xdx
=1/2∫{1/[(5/3)^x-1] - 1/[(5/3)^x 1]}×(5/3)^xdx
=1/2∫{1/[(5/3)^x-1] - 1/[(5/3)^x 1]}×d(5/3)^x/ln(5/3)
=1/2[1/ln(5/3)]×[ln|(5/3)^x-1|-ln|(5/3)^x 1| C
=1/2[1/ln(5/3)]×ln|[(5/3)^x-1]/[(5/3)^x 1]| C
=1/2[1/ln(5/3)]×ln|(5^x-3^x)/(5^x 3^x]| C