Integralen av $$$\ln\left(x\right) \cosh{\left(x \right)}$$$

Bestäm $$$\int \ln\left(x\right) \cosh{\left(x \right)}\, dx$$$.

För integralen $$$\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=\ln{\left(x \right)}$$$ och $$$\operatorname{dv}=\cosh{\left(x \right)} dx$$$.

Då gäller $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{\cosh{\left(x \right)} d x}=\sinh{\left(x \right)}$$$ (stegen kan ses »).

Alltså,

$${\color{red}{\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \sinh{\left(x \right)}-\int{\sinh{\left(x \right)} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\ln{\left(x \right)} \sinh{\left(x \right)} - \int{\frac{\sinh{\left(x \right)}}{x} d x}\right)}}$$

Denna integral (Hyperbolisk sinusintegral) har ingen sluten form:

$$\ln{\left(x \right)} \sinh{\left(x \right)} - {\color{red}{\int{\frac{\sinh{\left(x \right)}}{x} d x}}} = \ln{\left(x \right)} \sinh{\left(x \right)} - {\color{red}{\operatorname{Shi}{\left(x \right)}}}$$

Alltså,

$$\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x} = \ln{\left(x \right)} \sinh{\left(x \right)} - \operatorname{Shi}{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x} = \ln{\left(x \right)} \sinh{\left(x \right)} - \operatorname{Shi}{\left(x \right)}+C$$

$$$\int \ln\left(x\right) \cosh{\left(x \right)}\, dx = \left(\ln\left(x\right) \sinh{\left(x \right)} - \operatorname{Shi}{\left(x \right)}\right) + C$$$A