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

The integral of (9^x+81^x)/(1+81^x) is (arctan(9^x))/(2ln(3))+1/(4ln(3))ln|1+81^x|+C

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

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

\frac{arctan ( 9 ^{x} )}{2 ln ( 3 )} + \frac{1}{4 ln ( 3 )} ln ∣ 1 + 8 1^{x} ∣ + C

Solution steps

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

Expand

\frac{9 ^{x} + 8 1 ^{x}}{1 + 8 1 ^{x}} : \frac{9 ^{x}}{1 + 8 1 ^{x}} + \frac{8 1 ^{x}}{1 + 8 1 ^{x}}

Apply the Sum Rule:

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

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

\int \frac{9 ^{x}}{1 + 8 1 ^{x}} d x = \frac{arctan ( 9 ^{x} )}{2 ln ( 3 )}

\int \frac{8 1 ^{x}}{1 + 8 1 ^{x}} d x = \frac{1}{4 ln ( 3 )} ln ∣ 1 + 8 1^{x} ∣

= \frac{arctan ( 9 ^{x} )}{2 ln ( 3 )} + \frac{1}{4 ln ( 3 )} ln ∣ 1 + 8 1^{x} ∣

Add a constant to the solution

= \frac{arctan ( 9 ^{x} )}{2 ln ( 3 )} + \frac{1}{4 ln ( 3 )} ln ∣ 1 + 8 1^{x} ∣ + C

What is the integral of (9^x+81^x)/(1+81^x) ?
The integral of (9^x+81^x)/(1+81^x) is (arctan(9^x))/(2ln(3))+1/(4ln(3))ln|1+81^x|+C