Remark

If, in formula (27), one restricts the function $e^{a z}$ ($a$ complex) to the real axis, one again obtains, for $x$ real,

This formula allows us to calculate a primitive for each of the functions $e^{\alpha x} \cos \beta x$, $e^{\alpha x} \sin \beta x$ ($\alpha$ and $\beta$ real); indeed we have $e^{(\alpha + i \beta)x} = e^{\alpha x} \cos \beta x + i e^{\alpha x} \sin \beta x$, so, by (28) In the same way one reduces the evaluation of a primitive of $x^n e^{\alpha x} \cos \beta x$, or of $x^n e^{\alpha x} \sin \beta x$ (n an integer $> 0$) to that of a primitive of $x^n e^{(\alpha + i \beta)x}$; now, the formula for integration by parts of order $n + 1$ (II, p. 60, formula (11)) shows that a primitive of this last function is By Euler’s formulae one can on the other hand express every positive integral power of $\cos x$ or of $\sin x$ as a linear combination of exponentials $e^{ip x}$ ($p$ a positive or negative integer). By formula (28) one can thus express a primitive of a function of the form $x^n e^{\alpha x} (\cos \beta x)' (\sin \gamma x)^s$ as a linear combination of functions of the form $x^p e^{\alpha x} \cos \lambda x$ and $x^p e^{\alpha x} \sin \mu x$ and ($n, p, r, s$ integers, $\alpha, \beta, \gamma, \lambda, \mu$ real).