Integralna część e2xe^{2 x}

Kalkulator znajdzie całkę/pochodną e2xe^{2 x}, z pokazanymi krokami.

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Rozwiązanie

Let u=2xu=2 x.

Then du=(2x)dx=2dxdu=\left(2 x\right)^{\prime }dx = 2 dx (steps can be seen »), and we have that dx=du2dx = \frac{du}{2}.

The integral can be rewritten as

e2xdx=eu2du{\color{red}{\int{e^{2 x} d x}}} = {\color{red}{\int{\frac{e^{u}}{2} d u}}}

Apply the constant multiple rule cf(u)du=cf(u)du\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du with c=12c=\frac{1}{2} and f(u)=euf{\left(u \right)} = e^{u}:

eu2du=(eudu2){\color{red}{\int{\frac{e^{u}}{2} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{2}\right)}}

The integral of the exponential function is eudu=eu\int{e^{u} d u} = e^{u}:

eudu2=eu2\frac{{\color{red}{\int{e^{u} d u}}}}{2} = \frac{{\color{red}{e^{u}}}}{2}

Recall that u=2xu=2 x:

eu2=e(2x)2\frac{e^{{\color{red}{u}}}}{2} = \frac{e^{{\color{red}{\left(2 x\right)}}}}{2}

Dlatego,

e2xdx=e2x2\int{e^{2 x} d x} = \frac{e^{2 x}}{2}

Dodaj stałą całkowania:

e2xdx=e2x2+C\int{e^{2 x} d x} = \frac{e^{2 x}}{2}+C

Answer: e2xdx=e2x2+C\int{e^{2 x} d x}=\frac{e^{2 x}}{2}+C