integral of (15^x+30^x)/(5^x)

Lösung

\int \frac{1 5 ^{x} + 3 0 ^{x}}{5 ^{x}} d x

\frac{3 ^{x}}{ln ( 3 )} + \frac{6 ^{x}}{ln ( 6 )} + C

Schritte zur Lösung

\int \frac{1 5 ^{x} + 3 0 ^{x}}{5 ^{x}} d x

Multipliziere aus

\frac{1 5 ^{x} + 3 0 ^{x}}{5 ^{x}} : \frac{1 5 ^{x}}{5 ^{x}} + \frac{3 0 ^{x}}{5 ^{x}}

Wende die Summenregel an:

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

= \int \frac{1 5 ^{x}}{5 ^{x}} d x + \int \frac{3 0 ^{x}}{5 ^{x}} d x

\int \frac{1 5 ^{x}}{5 ^{x}} d x = \frac{3 ^{x}}{ln ( 3 )}

\int \frac{3 0 ^{x}}{5 ^{x}} d x = \frac{6 ^{x}}{ln ( 6 )}

= \frac{3 ^{x}}{ln ( 3 )} + \frac{6 ^{x}}{ln ( 6 )}

Füge eine Konstante zur Lösung hinzu

= \frac{3 ^{x}}{ln ( 3 )} + \frac{6 ^{x}}{ln ( 6 )} + C

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