What is the integral of (9^x)/(1+81^x) ?

The integral of (9^x)/(1+81^x) is (arctan(9^x))/(2ln(3))+C

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integral of (9^x)/(1+81^x)

\int \frac{9 ^{x}}{1 + 8 1 ^{x}} d x

\frac{arctan ( 9 ^{x} )}{2 ln ( 3 )} + C

Solution steps

\int \frac{9 ^{x}}{1 + 8 1 ^{x}} d x

Apply u-substitution

: \int \frac{1}{4 ln ( 3 ) \cdot 9 ^{\frac{l n ( u - 1 )}{4 l n ( 3 )}} u} d u

= \int \frac{1}{4 ln ( 3 ) \cdot 9 ^{\frac{l n ( u - 1 )}{4 l n ( 3 )}} u} d u

Take the constant out:

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

= \frac{1}{4 ln ( 3 )} \cdot \int \frac{1}{9 ^{\frac{l n ( u - 1 )}{4 l n ( 3 )}} u} d u

Apply u-substitution

: \int \frac{2}{v ^{2} + 1} d v

= \frac{1}{4 ln ( 3 )} \cdot \int \frac{2}{v ^{2} + 1} d v

Take the constant out:

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

= \frac{1}{4 ln ( 3 )} \cdot 2 \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{4 ln ( 3 )} \cdot 2 arctan (v)

Substitute back

= \frac{1}{4 ln ( 3 )} \cdot 2 arctan (9^{\frac{l n ( 1 + 8 1 ^{x} - 1 )}{4 l n ( 3 )}})

Simplify

\frac{1}{4 ln ( 3 )} \cdot 2 arctan (9^{\frac{l n ( 1 + 8 1 ^{x} - 1 )}{4 l n ( 3 )}}) : \frac{arctan ( 9 ^{x} )}{2 ln ( 3 )}

= \frac{arctan ( 9 ^{x} )}{2 ln ( 3 )}

Add a constant to the solution

= \frac{arctan ( 9 ^{x} )}{2 ln ( 3 )} + C

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What is the integral of (9^x)/(1+81^x) ?
The integral of (9^x)/(1+81^x) is (arctan(9^x))/(2ln(3))+C