Integraal van $$$\frac{e^{\frac{1}{x^{3}}}}{x^{4}}$$$

Bepaal $$$\int \frac{e^{\frac{1}{x^{3}}}}{x^{4}}\, dx$$$.

Zij $$$u=x^{3}$$$.

Dan $$$du=\left(x^{3}\right)^{\prime }dx = 3 x^{2} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$x^{2} dx = \frac{du}{3}$$$.

Dus,

$${\color{red}{\int{\frac{e^{\frac{1}{x^{3}}}}{x^{4}} d x}}} = {\color{red}{\int{\frac{e^{\frac{1}{u}}}{3 u^{2}} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{3}$$$ en $$$f{\left(u \right)} = \frac{e^{\frac{1}{u}}}{u^{2}}$$$:

$${\color{red}{\int{\frac{e^{\frac{1}{u}}}{3 u^{2}} d u}}} = {\color{red}{\left(\frac{\int{\frac{e^{\frac{1}{u}}}{u^{2}} d u}}{3}\right)}}$$

Zij $$$v=\frac{1}{u}$$$.

Dan $$$dv=\left(\frac{1}{u}\right)^{\prime }du = - \frac{1}{u^{2}} du$$$ (de stappen zijn te zien »), en dan geldt dat $$$\frac{du}{u^{2}} = - dv$$$.

De integraal kan worden herschreven als

$$\frac{{\color{red}{\int{\frac{e^{\frac{1}{u}}}{u^{2}} d u}}}}{3} = \frac{{\color{red}{\int{\left(- e^{v}\right)d v}}}}{3}$$

Pas de constante-veelvoudregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ toe met $$$c=-1$$$ en $$$f{\left(v \right)} = e^{v}$$$:

$$\frac{{\color{red}{\int{\left(- e^{v}\right)d v}}}}{3} = \frac{{\color{red}{\left(- \int{e^{v} d v}\right)}}}{3}$$

De integraal van de exponentiële functie is $$$\int{e^{v} d v} = e^{v}$$$:

$$- \frac{{\color{red}{\int{e^{v} d v}}}}{3} = - \frac{{\color{red}{e^{v}}}}{3}$$

We herinneren eraan dat $$$v=\frac{1}{u}$$$:

$$- \frac{e^{{\color{red}{v}}}}{3} = - \frac{e^{{\color{red}{\frac{1}{u}}}}}{3}$$

We herinneren eraan dat $$$u=x^{3}$$$:

$$- \frac{e^{{\color{red}{u}}^{-1}}}{3} = - \frac{e^{{\color{red}{x^{3}}}^{-1}}}{3}$$

Dus,

$$\int{\frac{e^{\frac{1}{x^{3}}}}{x^{4}} d x} = - \frac{e^{\frac{1}{x^{3}}}}{3}$$

Voeg de integratieconstante toe:

$$\int{\frac{e^{\frac{1}{x^{3}}}}{x^{4}} d x} = - \frac{e^{\frac{1}{x^{3}}}}{3}+C$$

$$$\int \frac{e^{\frac{1}{x^{3}}}}{x^{4}}\, dx = - \frac{e^{\frac{1}{x^{3}}}}{3} + C$$$A