integral of (e^{ax}-e^{bx})/(e^{cx)}

Lösung

\int \frac{e ^{a x} - e ^{b x}}{e ^{c x}} d x

\frac{1}{a - c} e^{a x - c x} - \frac{1}{b - c} e^{b x - c x} + C

Schritte zur Lösung

\int \frac{e ^{a x} - e ^{b x}}{e ^{c x}} d x

Multipliziere aus

\frac{e ^{a x} - e ^{b x}}{e ^{c x}} : e^{a x - c x} - e^{b x - c x}

= \int e^{a x - c x} - e^{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 e^{a x - c x} d x - \int e^{b x - c x} d x

\int e^{a x - c x} d x = \frac{1}{a - c} e^{a x - c x}

\int e^{b x - c x} d x = \frac{1}{b - c} e^{b x - c x}

= \frac{1}{a - c} e^{a x - c x} - \frac{1}{b - c} e^{b x - c x}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{a - c} e^{a x - c x} - \frac{1}{b - c} e^{b x - c x} + C

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

\int \frac{x ^{2} - 3 x + 3 x}{x ^{3}} d x

\int x (x^{2} + 5)^{9} d x

\int \frac{cos ^{3} ( 5 x )}{sin ^{4} ( 5 x )} d x

\int \frac{34}{x ^{2} x ^{2} + 49} d x