Page 244 -

Applied Mathematics, Calculus, and Differential Equations 237

( f (x) f (x) f (x) ... f (x)) dx
2
n
1
3
f (x) dx f (x) dx f (x) dx ... f (x) dx
n
2
1
3
IndOnite integral of variablł raised to
integer power
Let x be a variable. Let k be an integer wità the restriction that
k 1, and let c be the constant of integration. The following
formulł applies:

k
xSx (x k 1 /(k 1)) c

IndOnite integral of constanŁ raised to
variablł power
Let x be a variable. Let k be a constant such that k 0 and
k 1. Let c be the constant of integration. The following for-
mulł applies:

x
x
kdx (k /(ln k)) c

IndOnite integral of sinł wave

The antiderivative (indefinite integra„ of a sine wave is shown
in Fig. 3.19. The amplitude and displacement of the resultant
wave depend on the amplitude and frequency of the sine wave.

IndOnite integral of up-ramp wave
The antiderivative (indefinite integra„ of an up-ramp wave is
shown in Fig. 3.20˜ The magnitude of the resultant dependð on
the amplitude and frequency of the up-ramp wave.

IndOnite integral of down-ramp wave
The antiderivative (indefinite integra„ of a down-ramp wave is
shown in Fig. 3.21˜ The magnitude of the resultant dependð on
the amplitude and frequency of the down-ramp wave.