Question

`int (a^x - b^x)^2/(a^xb^x)dx` equals ______.

Options

• `(a/b)^x + 2x + c`

• `(b/a)^x + 2x + c`

• `(a/b)^x - 2x + c`

• `1/(ln(a/b))[(a^(2x) - b^(2x))/(a^xb^x)] - 2x + c`

Answer 1

`int (a^x - b^x)^2/(a^xb^x)dx` equals `underlinebb(1/(ln(a/b))[(a^(2x) - b^(2x))/(a^xb^x)] - 2x + c`.

Explanation:

`int (a^x - b^x)^2/(a^xb^x)dx`

= `int(a^x/b^x + b^x/a^x - 2)dx`

= `int[(a/b)^x + (b/a)^x - 2]dx`

= `(a/b)^x/(ln(a/b)) + (b/a)^x/(ln(b/a)) - 2x + c`

= `1/(ln(a/b))[(a/b)^2 - (b/a)^2] - 2x + c`

= `1/(ln(a/b))[(a^(2x) - b^(2x))/(a^xb^x)] - 2x + c`