Integraal van $$$5^{x^{2}} x$$$

Bepaal $$$\int 5^{x^{2}} x\, dx$$$.

Zij $$$u=x^{2}$$$.

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

Dus,

$${\color{red}{\int{5^{x^{2}} x d x}}} = {\color{red}{\int{\frac{5^{u}}{2} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = 5^{u}$$$:

$${\color{red}{\int{\frac{5^{u}}{2} d u}}} = {\color{red}{\left(\frac{\int{5^{u} d u}}{2}\right)}}$$

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

$$\frac{{\color{red}{\int{5^{u} d u}}}}{2} = \frac{{\color{red}{\frac{5^{u}}{\ln{\left(5 \right)}}}}}{2}$$

We herinneren eraan dat $$$u=x^{2}$$$:

$$\frac{5^{{\color{red}{u}}}}{2 \ln{\left(5 \right)}} = \frac{5^{{\color{red}{x^{2}}}}}{2 \ln{\left(5 \right)}}$$

Dus,

$$\int{5^{x^{2}} x d x} = \frac{5^{x^{2}}}{2 \ln{\left(5 \right)}}$$

Voeg de integratieconstante toe:

$$\int{5^{x^{2}} x d x} = \frac{5^{x^{2}}}{2 \ln{\left(5 \right)}}+C$$

$$$\int 5^{x^{2}} x\, dx = \frac{5^{x^{2}}}{2 \ln\left(5\right)} + C$$$A