Question

Integrate the following functions with respect to x :

`(8^(1 + x) + 4^(1 - x))/2^x`

Answer 1

`int ((8^(1 + x) + 4^(1 - x))/2^x) "d"x`

= `int ((8^(1 + x))/2^x + (4^(1- x))/2^x) "d"x`

= `int ((8^1 * 8^x)/2^x + (4^1 * 4^-x)/2^2x) "d"x`

= `int (8* 8^x)/2^x  "d"x + int (4* 4^-x)/2^x * "d"x`

= `8 int (2^3)^x/2^x  "d"x + 4int (2^2)^x/2^x * "d"x`

= `8int 2^(3x)/2^x "d"x + 4int 2^(-2x)/2^x * "d"x`

= `8int 2^(3x) 2^(- x) * "d"x + 4int 2^(- 2x) 2^(- x) * "d"x`

= `8int 2^(3x - x) * "d"x + 4int 2^(- 2x - x) * "d"x`

= `8int 2^(2x) * "d"x + 4int 2^(- 3x) * "d"x`

= `8 xx 1/2 xx 2^(2x)/log 2 + 4 xx 1/(-3) xx 2^(- 3x)/log2 + "c"`

= `4 xx 2^(2x)/log2 - 4/3 xx 2^(- 3x)/log2 + "c"`

= `(2^(2x) xx 2^(2x))/log2 - (2^2 xx 2^(- 3x))/(3 log 2) + "c"`

= `2^(2x + 2)/log2 - 2^(2 - 3x)/(3log2) + "c"`