Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Errata

Errata for 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables

Chapter $2$: Special Products and Factors: $2.22$: Difference of Two Odd Powers:

$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \paren {x^{n - 1} + x^{n - 2} y + x^{n - 3} y^2 + \dotsb} \paren {x^{n - 1} - x^{n - 2} y + x^{n - 3} y^2 - \dotsb}$

Chapter $5$: Trigonometric Functions: Relationships between Sides and Angles of a Spherical Triangle: $5.99$: Cosine of Half Angle for Spherical Triangles:

$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - c} } {\sin b \sin c} }$

where $s = \dfrac {a + b + c} 2$.

Chapter $14$: Indefinite Integrals: General Rules of Integration: $14.30$: Primitive of $\csch u$:

$\ds \int \csch u \rd u = -\map {\coth^{-1} } {e^u} + C$

Chapter $14$: Indefinite Integrals: Integrals involving $a x + b$: $14.73$: Primitive of $\dfrac 1 {\paren {a x + b}^3}$:

$\ds \int \frac {\d x} {\paren {a x + b}^3} = -\frac 1 {2 \paren {a x + b}^2} + C$

Chapter $14$: Indefinite Integrals: Integrals involving $a x + b$: $14.79$: Primitive of $\dfrac 1 {x^3 \paren {a x + b}^3}$:

$\ds \int \frac {\d x} {x^3 \paren {a x + b}^3} = \frac {a^4 x^2} {2 b^5 \paren {a x + b}^2} - \frac {4 a^3 x} {b^5 \paren {a x + b} } - \frac {\paren {a x + b}^2} {2 b^5 x^2} + \frac {6 a^2} {b^5} \ln \size {\frac x {a x + b} } + C$

Chapter $14$: Indefinite Integrals: Integrals involving $\sqrt {a x + b}$ and $\sqrt {p x + q}$: $14.120$: Primitive of $\dfrac 1 {\sqrt {\paren {a x + b} \paren {p x + q} } }$:

$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases}

\dfrac 2 {\sqrt {a p} } \map \ln {\sqrt {p \paren {a x + b} } + \sqrt {a \paren {p x + q} } } \\ \dfrac 2 {\sqrt {-a p} } \tan^{-1} \sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } \end{cases}$

Chapter $14$: Indefinite Integrals: Integrals involving $\cosh a x$: $14.584$: Primitive of $\dfrac {\d x} {p^2 + q^2 \cosh^2 a x}$:

$\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x} = \begin {cases}

\dfrac 1 {2 a p \sqrt {p^2 + q^2} } \map \ln {\dfrac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } } \\ \dfrac 1 {a p \sqrt {p^2 + q^2} } \arctan \dfrac {p \tanh a x} {\sqrt {p^2 + q^2} } \end {cases}$

Chapter $14$: Indefinite Integrals: Integrals involving Inverse Hyperbolic Functions: $14.655$: Primitive of $\dfrac 1 {x^2} \cosh^{-1} \dfrac x a$:

$\ds \int \frac {\map {\cosh^{-1} } {x / a} } {x^2} \rd x = \dfrac {-\map {\cosh^{-1} } {x / a} } x \mp \dfrac 1 a \map \ln {\dfrac {a + \sqrt {x^2 + a^2} } x}$ $\sqbrk {- \text { if } \map {\cosh^{-1} } {x / a} > 0, + \text { if } \map {\cosh^{-1} } {x / a} < 0}$

Chapter $15$: Indefinite Integrals: Definite Integrals involving Trigonometric Functions: $15.34$: Definite integral $\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x$:

$\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} 0 & : p > q > 0 \\

\dfrac \pi 2 & : 0 < p < q \\ \dfrac \pi 4 & : p = q > 0 \end {cases}$

Chapter $20$: Taylor Series: Series for Hyperbolic Functions: $20.36$: Power Series Expansion for Hyperbolic Cotangent Function:

$\coth x = \dfrac 1 x + \dfrac x 3 - \dfrac {x^3} {45} + \dfrac {2 x^5} {945} + \cdots \dfrac {\paren {-1}^n 2^{2 n} B_n x^{2 n - 1} } {\paren {2 n}!} + \cdots \quad 0 < \size x < \pi$