C.3 Hyperbolic Functions

K_1 \cos x + K_2 \sin x = \sqrt{K_1^2 + K_2^2} \cos \left(x + \tan^{-1} \frac{-K_2}{K_1}\right)

e^{jx} = \cos x + j \sin x

(Euler’s formula)

\cos x = \frac{e^{jx} + e^{-jx}}{2}

\sin x = \frac{e^{jx} - e^{-jx}}{2j}

1 \text{ rad} = 57.296^{\circ}

\sinh x = \frac{1}{2} (e^x - e^{-x})

\cosh x = \frac{1}{2} (e^x + e^{-x})

\tanh x = \frac{\sinh x}{\cosh x}

\coth x = \frac{1}{\tanh x}

\operatorname{csch} x = \frac{1}{\sinh x}

\operatorname{sech} x = \frac{1}{\cosh x}

sinh(x ± y) = sinh x cosh y ± cosh x sinh y cosh(x ± y) = cosh x cosh y ± sinh x sinh y

C.4 Derivatives

If U = U(x), V = V(x), and a = constant,

\frac{d}{dx}(aU) = a\frac{dU}{dx}

\frac{d}{dx}(UV) = U\frac{dV}{dx} + V\frac{dU}{dx}

\frac{d}{dx}\left(\frac{U}{V}\right) = \frac{V\frac{dU}{dx} - U\frac{dV}{dx}}{V^2}

\frac{d}{dx}(aU^n) = naU^{n-1}

\frac{d}{dx}(a^U) = a^U \ln a \frac{dU}{dx}

\frac{d}{dx}(e^U) = e^U \frac{dU}{dx}

\frac{d}{dx}(\sin U) = \cos U \frac{dU}{dx}

\frac{d}{dx}(\cos U) = -\sin U \frac{dU}{dx}

U = U(x)

, $V = V(x)$ , and $a = \text{constant}$ ,
\n

\int a \, dx = ax + C

\int U \, dV = UV - \int V \, dU \qquad \text{(integration by parts)}

\int U^n \, dU = \frac{U^{n+1}}{n+1} + C, \qquad n \neq 1

\int \frac{dU}{U} = \ln U + C

\int a^U \, dU = \frac{a^U}{\ln a} + C, \qquad a > 0, a \neq 1

\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C

\int xe^{ax} \, dx = \frac{e^{ax}}{a^2} (ax - 1) + C

\int x^2 e^{ax} \, dx = \frac{e^{ax}}{a^3} (a^2 x^2 - 2ax + 2) + C

\int \ln x \, dx = x \ln x - x + C

\int \sin ax \, dx = -\frac{1}{a} \cos ax + C

\int \cos ax \, dx = \frac{1}{a} \sin ax + C

\int \cos^2 ax \, dx = \frac{x}{2} - \frac{\sin 2ax}{4a} + C

\int \cos^2 ax \, dx = \frac{x}{2} + \frac{\sin 2ax}{4a} + C

\int x \sin ax \, dx = \frac{1}{a^2} (\sin ax - ax \cos ax) + C

\int x \cos ax \, dx = \frac{1}{a^2} (\cos ax + ax \sin ax) + C

\int x^2 \sin ax \, dx = \frac{1}{a^3} (2ax \sin ax + 2 \cos ax - a^2x^2 \cos ax) + C

\int x^2 \cos ax \, dx = \frac{1}{a^3} (2ax \cos ax - 2 \sin ax + a^2x^2 \sin ax) + C

\int e^{ax} \sin bx \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx) + C

\int e^{ax} \cos bx \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos bx + b \sin bx) + C

\int \sin ax \sin bx \, dx = \frac{\sin(a - b)x}{2(a - b)} - \frac{\sin(a + b)x}{2(a + b)} + C, \quad a^2 \neq b^2

\int \sin ax \cos bx \, dx = -\frac{\cos(a - b)x}{2(a - b)} - \frac{\cos(a + b)x}{2(a + b)} + C, \quad a^2 \neq b^2

\int \cos ax \cos bx \, dx = \frac{\sin(a - b)x}{2(a - b)} + \frac{\sin(a + b)x}{2(a + b)} + C, \quad a^2 \neq b^2

\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + C

\int \frac{x^2 dx}{a^2 + x^2} = \frac{1}{2a^2} \left( \frac{x}{x^2 + a^2} + \frac{1}{a} \tan^{-1} \frac{x}{a} \right) + C

If m and n are integers,

\int_{0}^{2\pi} \sin ax \, dx = 0

\int_{0}^{2\pi} \cos ax \, dx = 0

\int_{0}^{\pi} \sin^{2} ax \, dx = \int_{0}^{\pi} \cos^{2} ax \, dx = \frac{\pi}{2}

\int_{0}^{\pi} \sin mx \sin nx \, dx = \int_{0}^{\pi} \cos mx \cos nx \, dx = 0, \quad m \neq n

\int_{0}^{\pi} \sin mx \cos nx \, dx = \begin{cases} 0, & m + n = \text{even} \\ \frac{2m}{m^{2} - n^{2}}, & m + n = \text{odd} \end{cases}

\int_{0}^{2\pi} \sin mx \sin nx \, dx = \int_{-\pi}^{\pi} \sin mx \sin nx \, dx = \begin{cases} 0, & m \neq n \\ \pi, & m = n \end{cases}

\int_0^\infty \frac{\sin ax}{x} dx = \begin{cases} \frac{\pi}{2}, & a > 0 \\ 0, & a = 0 \\ -\frac{\pi}{2}, & a < 0 \end{cases}

If f(0) = 0 = h(0), then

\lim_{x \to 0} \frac{f(x)}{h(x)} = \lim_{x \to 0} \frac{f'(x)}{h'(x)}

where the prime indicates differentiation.

1.1 (a) −103.84 mC, (b) −198.65 mC, (c) −3.941 C, (d) −26.08 C
1.3 (a) 3t + 1 C, (b) t 2 + 5t mC, (c) 2 sin(10t + π∕6) + 1 μC, (d) −e⁻30*^t* [0.16 cos 40t + 0.12 sin 40t] C
1.5 25 C

1.7

i = \frac{dq}{dt} = \begin{cases} 10 \text{ A}, & 0 < t < 1 \\ -20 \text{ A}, & 1 < t < 2 \\ 0 \text{ A}, & 2 < t < 3 \\ 10 \text{ A}, & 3 < t < 4 \end{cases}

1.27 (a) 43.2 kC, (b) 475.2 kJ, (c) 1.188 cents

See the sketch in Fig. D.1.

Figure D.1

For Prob. 1.7.