What is the integral of ((8^x+1))/(2^x+1) ?

The integral of ((8^x+1))/(2^x+1) is (2^{2x-1}-2^x+ln(2)x)/(ln(2))+C

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integral of ((8^x+1))/(2^x+1)

\int \frac{( 8 ^{x} + 1 )}{2 ^{x} + 1} d x

\frac{2 ^{2 x - 1} - 2 ^{x} + ln ( 2 ) x}{ln ( 2 )} + C

Solution steps

\int \frac{8 ^{x} + 1}{2 ^{x} + 1} d x

Apply u-substitution

: \int \frac{u + 1}{3 ln ( 2 ) u ( 2 ^{\frac{l n ( u )}{3 l n ( 2 )}} + 1 )} d u

= \int \frac{u + 1}{3 ln ( 2 ) u ( 2 ^{\frac{l n ( u )}{3 l n ( 2 )}} + 1 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3 ln ( 2 )} \cdot \int \frac{u + 1}{u ( 2 ^{\frac{l n ( u )}{3 l n ( 2 )}} + 1 )} d u

Apply u-substitution

: \int \frac{3 ( v ^{2} - 3 v + 3 )}{v - 1} d v

= \frac{1}{3 ln ( 2 )} \cdot \int \frac{3 ( v ^{2} - 3 v + 3 )}{v - 1} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3 ln ( 2 )} \cdot 3 \cdot \int \frac{v ^{2} - 3 v + 3}{v - 1} d v

Expand

\frac{v ^{2} - 3 v + 3}{v - 1} : \frac{v ^{2}}{v - 1} - \frac{3 v}{v - 1} + \frac{3}{v - 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3 ln ( 2 )} \cdot 3 (\int \frac{v ^{2}}{v - 1} d v - \int \frac{3 v}{v - 1} d v + \int \frac{3}{v - 1} d v)

\int \frac{v ^{2}}{v - 1} d v = \frac{( v - 1 ) ^{2}}{2} + 2 (v - 1) + ln ∣ v - 1 ∣

\int \frac{3 v}{v - 1} d v = 3 (v - 1 + ln ∣ v - 1 ∣)

\int \frac{3}{v - 1} d v = 3 ln ∣ v - 1 ∣

= \frac{1}{3 ln ( 2 )} \cdot 3 (\frac{( v - 1 ) ^{2}}{2} + 2 (v - 1) + ln ∣ v - 1 ∣ - 3 (v - 1 + ln ∣ v - 1 ∣) + 3 ln ∣ v - 1 ∣)

Substitute back

= \frac{1}{3 ln ( 2 )} \cdot 3 \frac{( 2 ^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 ) ^{2}}{2} + 2 (2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1) + ln 2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 - 3 (2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 + ln 2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1) + 3 ln 2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1

Simplify

\frac{1}{3 ln ( 2 )} \cdot 3 \frac{( 2 ^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 ) ^{2}}{2} + 2 (2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1) + ln 2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 - 3 (2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 + ln 2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1) + 3 ln 2^{\frac{l n ( 8 ^{x} )}{3 l n ( 2 )}} + 1 - 1 : \frac{2 ^{2 x - 1} - 2 ^{x} + ln ( 2 ) x}{ln ( 2 )}

= \frac{2 ^{2 x - 1} - 2 ^{x} + ln ( 2 ) x}{ln ( 2 )}

Add a constant to the solution

= \frac{2 ^{2 x - 1} - 2 ^{x} + ln ( 2 ) x}{ln ( 2 )} + C

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What is the integral of ((8^x+1))/(2^x+1) ?
The integral of ((8^x+1))/(2^x+1) is (2^{2x-1}-2^x+ln(2)x)/(ln(2))+C