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

解

\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

解答ステップ

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

拡張

\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}}

総和規則を適用する:

\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} ∣

定数を解答に追加する

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