Integrals of Gradshteyn and Ryzhik: 2.671 and 2.672 – Combinations of trigonometric and hyperbolic functions

Here, integration by parts is used twice for each integral. We repeatedly use the following:
\begin{equation}
u=\sinh(ax+b) \quad du=a\,\cosh(ax+b) \quad \mathrm{and} \quad u=\cosh(ax+b) \quad du=a\,\sinh(ax+b)
\end{equation}
\begin{equation}
dv=\sin(cx+d) \quad v=-\frac{\cos(cx+d)}{c} \quad \mathrm{and} \quad dv=\cos(cx+d) \quad v=\frac{\sin(cx+d)}{c}
\end{equation}

2.671.1

\begin{equation}
\int\sinh(ax+b)\sin(cx+d) dx = -\frac{\sinh(ax+b)\cos(cx+d)}{c} + \frac{a}{c}\int\cosh(ax+b)\cos(cx+d) dx
\end{equation}
\begin{equation}
= -\frac{\sinh(ax+b)\cos(cx+d)}{c} + \frac{a}{c} \left[ \frac{\cosh(ax+b)\sin(cx+d)}{c}\,-\frac{a}{c}\int\sinh(ax+b)\sin(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\cosh(ax+b)\sin(cx+d)\,-c\,\sinh(ax+b)\cos(cx+d)}{a^2 + c^2}
\end{equation}

2.671.2

\begin{equation}
\int\sinh(ax+b)\cos(cx+d) dx = \frac{\sinh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c}\int\cosh(ax+b)\sin(cx+d) dx
\end{equation}
\begin{equation}
= \frac{\sinh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c} \left[-\frac{\cosh(ax+b)\cos(cx+d)}{c}+\frac{a}{c}\int\sinh(ax+b)\cos(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\cosh(ax+b)\cos(cx+d) + c\,\sinh(ax+b)\sin(cx+d)}{a^2 + c^2}
\end{equation}

2.671.3

\begin{equation}
\int\cosh(ax+b)\sin(cx+d) dx = -\frac{\cosh(ax+b)\cos(cx+d)}{c} + \frac{a}{c}\int\sinh(ax+b)\cos(cx+d) dx
\end{equation}
\begin{equation}
= -\frac{\cosh(ax+b)\cos(cx+d)}{c} + \frac{a}{c} \left[ \frac{\sinh(ax+b)\sin(cx+d)}{c}\,-\frac{a}{c}\int\cosh(ax+b)\sin(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\sinh(ax+b)\sin(cx+d)\,-c\,\cosh(ax+b)\cos(cx+d)}{a^2 + c^2}
\end{equation}

2.671.4

\begin{equation}
\int\cosh(ax+b)\cos(cx+d) dx = \frac{\cosh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c}\int\sinh(ax+b)\sin(cx+d) dx
\end{equation}
\begin{equation}
= \frac{\cosh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c} \left[-\frac{\sinh(ax+b)\cos(cx+d)}{c}+\frac{a}{c}\int\cosh(ax+b)\cos(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\sinh(ax+b)\cos(cx+d) + c\,\cosh(ax+b)\sin(cx+d)}{a^2 + c^2}
\end{equation}

2.672.1

Let \(a=c=1,\,b=d=0\) in 2.671.1
\begin{equation}
\int\sinh(x)\sin(x) dx = \frac{1}{2}[\cosh(x)\sin(x)\,- \sinh(x)\cos(x)]
\end{equation}

2.672.2

Let \(a=c=1,\,b=d=0\) in 2.671.2
\begin{equation}
\int\sinh(x)\cos(x) dx = \frac{1}{2}[\sinh(x)\sin(x)+ \cosh(x)\cos(x)]
\end{equation}

2.672.3

Let \(a=c=1,\,b=d=0\) in 2.671.3
\begin{equation}
\int\cosh(x)\sin(x) dx = \frac{1}{2}[\sinh(x)\sin(x)\,- \cosh(x)\cos(x)]
\end{equation}

2.672.4

Let \(a=c=1,\,b=d=0\) in 2.671.4
\begin{equation}
\int\cosh(x)\cos(x) dx = \frac{1}{2}[\cosh(x)\sin(x)+ \sinh(x)\cos(x)]
\end{equation}

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