integral of (a^x+b^x)/(c^x)

Lösung

\int \frac{a ^{x} + b ^{x}}{c ^{x}} d x

\frac{a ^{x}}{c ^{x} ln ( \frac{a}{c} )} + \frac{b ^{x}}{c ^{x} ln ( \frac{b}{c} )} + C

Schritte zur Lösung

\int \frac{a ^{x} + b ^{x}}{c ^{x}} d x

Multipliziere aus

\frac{a ^{x} + b ^{x}}{c ^{x}} : \frac{a ^{x}}{c ^{x}} + \frac{b ^{x}}{c ^{x}}

= \int \frac{a ^{x}}{c ^{x}} + \frac{b ^{x}}{c ^{x}} d 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{a ^{x}}{c ^{x}} d x + \int \frac{b ^{x}}{c ^{x}} d x

\int \frac{a ^{x}}{c ^{x}} d x = \frac{a ^{x}}{c ^{x} ln ( \frac{a}{c} )}

\int \frac{b ^{x}}{c ^{x}} d x = \frac{b ^{x}}{c ^{x} ln ( \frac{b}{c} )}

= \frac{a ^{x}}{c ^{x} ln ( \frac{a}{c} )} + \frac{b ^{x}}{c ^{x} ln ( \frac{b}{c} )}

Füge eine Konstante zur Lösung hinzu

= \frac{a ^{x}}{c ^{x} ln ( \frac{a}{c} )} + \frac{b ^{x}}{c ^{x} ln ( \frac{b}{c} )} + C

\int \frac{1}{x ^{3} + 6 x ^{2} + 9 x} d x

\int cos (x) cos (3 x) cos (5 x) d x

\int \frac{1}{36 + ( x - 4 ) ^{2}} d x

\int 3 x^{2} (x^{3} - 3)^{4} d x

\int 8 x ln (6 x) d x