What is the integral of (3^x)/(9^x+3) ?

The integral of (3^x)/(9^x+3) is (arctan(3^{1/2 (2x-1)}))/(sqrt(3)ln(3))+C

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integral of (3^x)/(9^x+3)

\int \frac{3 ^{x}}{9 ^{x} + 3} d x

\frac{arctan ( 3 ^{\frac{1}{2} (2 x - 1)} )}{3 ln ( 3 )} + C

Solution steps

\int \frac{3 ^{x}}{9 ^{x} + 3} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integral Substitution

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

Substitute back

= \frac{1}{ln ( 3 )} \cdot \frac{1}{3} arctan (\frac{3 ^{x}}{3})

Simplify

\frac{1}{ln ( 3 )} \cdot \frac{1}{3} arctan (\frac{3 ^{x}}{3}) : \frac{arctan ( 3 ^{\frac{1}{2} (2 x - 1)} )}{3 ln ( 3 )}

= \frac{arctan ( 3 ^{\frac{1}{2} (2 x - 1)} )}{3 ln ( 3 )}

Add a constant to the solution

= \frac{arctan ( 3 ^{\frac{1}{2} (2 x - 1)} )}{3 ln ( 3 )} + C

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