Integralen av $$$- 5^{u}$$$

Bestäm $$$\int \left(- 5^{u}\right)\, du$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=-1$$$ och $$$f{\left(u \right)} = 5^{u}$$$:

$${\color{red}{\int{\left(- 5^{u}\right)d u}}} = {\color{red}{\left(- \int{5^{u} d u}\right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=5$$$:

$$- {\color{red}{\int{5^{u} d u}}} = - {\color{red}{\frac{5^{u}}{\ln{\left(5 \right)}}}}$$

Alltså,

$$\int{\left(- 5^{u}\right)d u} = - \frac{5^{u}}{\ln{\left(5 \right)}}$$

Lägg till integrationskonstanten:

$$\int{\left(- 5^{u}\right)d u} = - \frac{5^{u}}{\ln{\left(5 \right)}}+C$$

$$$\int \left(- 5^{u}\right)\, du = - \frac{5^{u}}{\ln\left(5\right)} + C$$$A