Advertisements
Advertisements
Question
Integrate the following functions with respect to x :
`(8^(1 + x) + 4^(1 - x))/2^x`
Solution
`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"`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int1/(x(3+logx))dx`
Integrate the following functions with respect to x :
cot2x + tan2x
Integrate the following functions with respect to x :
`(3 + 4cosx)/(sin^2x)`
Integrate the following functions with respect to x :
`(sin4x)/sinx`
Integrate the following functions with respect to x :
`1/((x - 1)(x + 2)^2`
Integrate the following with respect to x :
`cot x/(log(sin x))`
Integrate the following with respect to x :
sin5x cos3x
Integrate the following with respect to x:
27x2e3x
Integrate the following with respect to x:
x3 sin x
Integrate the following with respect to x:
`(x sin^-1 x)/sqrt(1 - x^2)`
Find the integrals of the following:
`1/(25 - 4x^2)`
Find the integrals of the following:
`1/sqrt((2 + x)^2 - 1)`
Integrate the following functions with respect to x:
`sqrt((x + 1)^2 - 4)`
Choose the correct alternative:
`int sqrt(tanx)/(sin2x) "d"x` is
Choose the correct alternative:
`int ("e"^(6 log x) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x` is
Choose the correct alternative:
`int secx/sqrt(cos2x) "d"x` is
Choose the correct alternative:
`int x^2 "e"^(x/2) "d"x` is
Choose the correct alternative:
`int 1/(x sqrt(log x)^2 - 5) "d"x` is
