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:

2.671.1

\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

= -\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]

= \frac{a\,\cosh(ax+b)\sin(cx+d)\,-c\,\sinh(ax+b)\cos(cx+d)}{a^2 + c^2}

2.671.2

\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

= \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]

= \frac{a\,\cosh(ax+b)\cos(cx+d) + c\,\sinh(ax+b)\sin(cx+d)}{a^2 + c^2}

2.671.3

\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

= -\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]

= \frac{a\,\sinh(ax+b)\sin(cx+d)\,-c\,\cosh(ax+b)\cos(cx+d)}{a^2 + c^2}

2.671.4

\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

= \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]

= \frac{a\,\sinh(ax+b)\cos(cx+d) + c\,\cosh(ax+b)\sin(cx+d)}{a^2 + c^2}

2.672.1

Let $$a=c=1,\,b=d=0$$ in 2.671.1

\int\sinh(x)\sin(x) dx = \frac{1}{2}[\cosh(x)\sin(x)\,- \sinh(x)\cos(x)]

2.672.2

Let $$a=c=1,\,b=d=0$$ in 2.671.2

\int\sinh(x)\cos(x) dx = \frac{1}{2}[\sinh(x)\sin(x)+ \cosh(x)\cos(x)]

2.672.3

Let $$a=c=1,\,b=d=0$$ in 2.671.3

\int\cosh(x)\sin(x) dx = \frac{1}{2}[\sinh(x)\sin(x)\,- \cosh(x)\cos(x)]

2.672.4

Let $$a=c=1,\,b=d=0$$ in 2.671.4

\int\cosh(x)\cos(x) dx = \frac{1}{2}[\cosh(x)\sin(x)+ \sinh(x)\cos(x)]