DownloadsDownloadAppendix A: Mathematical FormulasThe Project Elara contributorsAlgebraic identities¶Basic identities¶x+y=y+xx + y = y + xx+y=y+x(x+y)+z=x+(y+z)(x + y) + z = x + (y + z)(x+y)+z=x+(y+z)1⋅x=x1 \cdot x = x1⋅x=x−x=x⋅(−1)-x = x \cdot (-1)−x=x⋅(−1)x⋅y=y⋅xx \cdot y = y \cdot xx⋅y=y⋅x(x⋅y)⋅z=x⋅(y⋅z)(x\cdot y)\cdot z = x \cdot (y \cdot z)(x⋅y)⋅z=x⋅(y⋅z)x÷y=xyx \div y = \frac{x}{y}x÷y=yxTrigonometric identities¶Exponential and logarithmic identities¶List of useful derivatives¶List of useful integrals¶∫k dx=kx+C\int k~dx = kx + C∫k dx=kx+C(1)∫xndx=1n+1xn+1+C,n≠−1\int x^n dx = \frac{1}{n +1} x^{n + 1} + C, \quad n \neq -1∫xndx=n+11xn+1+C,n=−1(2)∫(ax+b)ndx=(ax+b)n+1(n+1)a+C,n≠1\int (ax + b)^n dx = \frac{\left(a x + b\right)^{n + 1}}{(n + 1)a} + C, \quad n \neq 1∫(ax+b)ndx=(n+1)a(ax+b)n+1+C,n=1(3)∫ax2+bx dx=23b(bx+c)3/2,b≠0\int \sqrt{ ax^2 + bx }~dx = \frac{2}{3b}(bx + c)^{3/2}, \quad b \neq 0∫ax2+bx dx=3b2(bx+c)3/2,b=0(4)∫ax2+bx+c dx={a(2ax+b)ax2+bx+c+(4ac−b2)ln(2aax2+bx+c+2ax+b)4a32for 4c−b2a≠0(2ax+b)(2a(2ax+b)2aax2+bx+c+(4ac−b2)ln(2ax+b2a))8∣2ax+b∣otherwise\int \sqrt{ ax^2 + bx + c }~dx =\begin{cases} \frac{\sqrt{a} \left(2 a x + b\right) \sqrt{a x^{2} + b x + c} + \left(4 a c - b^{2}\right) \ln{\left(2 \sqrt{a} \sqrt{a x^{2} + b x + c} + 2 a x + b \right)}}{4 a^{\frac{3}{2}}} & \text{for}\: 4 c - \frac{b^{2}}{a} \neq 0 \\ \frac{\left(2 a x + b\right) \left(2 a \sqrt{\frac{\left(2 a x + b\right)^{2}}{a}} \sqrt{a x^{2} + b x + c} + \left(4 a c - b^{2}\right) \ln{\left(\frac{2 a x + b}{2 a} \right)}\right)}{8 |2ax + b|} & \text{otherwise} \end{cases}∫ax2+bx+c dx=⎩⎨⎧4a23a(2ax+b)ax2+bx+c+(4ac−b2)ln(2aax2+bx+c+2ax+b)8∣2ax+b∣(2ax+b)(2aa(2ax+b)2ax2+bx+c+(4ac−b2)ln(2a2ax+b))for4c−ab2=0otherwise(5)∫dxax+b=1aln∣ax+b∣+C\int \frac{dx}{ax + b} = \frac{1}{a} \ln |ax + b| + C∫ax+bdx=a1ln∣ax+b∣+C(6)∫dxax+b=2ax+ba+C\int \frac{dx}{\sqrt{ ax + b }} = \frac{2 \sqrt{a x + b}}{a} + C∫ax+bdx=a2ax+b+C(7)∫dx(ax+b)3/2=−2aax+b+C\int \frac{dx}{(ax + b)^{3/2}} = - \frac{2}{a \sqrt{a x + b}} + C∫(ax+b)3/2dx=−aax+b2+C(8)∫dx(ax+b)5/2=−23a(ax+b)3/2+C\int \frac{dx}{(ax + b)^{5/2}} = - \frac{2}{3 a \left(a x + b\right)^{3 / 2}} + C∫(ax+b)5/2dx=−3a(ax+b)3/22+C(9)∫cos(ax)dx=1asin(ax)+C\int \cos(a x) dx = \frac{1}{a} \sin(ax) + C∫cos(ax)dx=a1sin(ax)+C(10)∫sin(ax)dx=−1acos(ax)+C\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + C∫sin(ax)dx=−a1cos(ax)+C(11)∫tan(ax)dx=−1aln∣cos(ax)∣+C,a≠0\int \tan(ax) dx = - \frac{1}{a} \ln|\cos(ax) | + C, \quad a \neq 0∫tan(ax)dx=−a1ln∣cos(ax)∣+C,a=0(12)∫sec(ax)tan(ax)dx=1asec(ax)+C\int \sec(ax) \tan(ax)dx = \frac{1}{a}\sec(ax) + C∫sec(ax)tan(ax)dx=a1sec(ax)+C(13)∫csc(ax)cot(ax)dx=−1acsc(ax)+C\int \csc(ax) \cot(ax)dx = -\frac{1}{a} \csc(ax) + C∫csc(ax)cot(ax)dx=−a1csc(ax)+C(14)∫sec2(ax)dx=1atan(ax)+C\int \sec^2(ax) dx = \frac{1}{a} \tan (ax) + C∫sec2(ax)dx=a1tan(ax)+C(15)∫0asin(mπxa)sin(nπxa) dx={a2,m=n0,m≠n\int_0^a \sin \left(\frac{m\pi x}{a}\right) \sin \left(\frac{n\pi x}{a}\right)\,dx = \begin{cases} \frac{a}{2}, & m = n \\ 0, & m \neq n \end{cases}∫0asin(amπx)sin(anπx)dx={2a,0,m=nm=n(16)∫0acos(mπxa)cos(nπxa) dx={a,m=n=0a2,m=n≠00,m≠n\int_0^a \cos \left(\frac{m\pi x}{a}\right) \cos \left(\frac{n\pi x}{a}\right)\,dx = \begin{cases} a, & m = n = 0 \\ \frac{a}{2}, & m = n \neq 0 \\ 0, & m \neq n \end{cases}∫0acos(amπx)cos(anπx)dx=⎩⎨⎧a,2a,0,m=n=0m=n=0m=n(17)∫−aasin(mπxa)cos(nπxa) dx=0\int_{-a}^a \sin \left(\frac{m\pi x}{a}\right) \cos \left(\frac{n\pi x}{a}\right)\,dx = 0∫−aasin(amπx)cos(anπx)dx=0(18)∫xsin(ax)dx=−xcos(ax)a+sin(ax)a2+C\int x \sin(ax) dx = - \frac{x \cos{\left(a x \right)}}{a} + \frac{\sin{\left(a x \right)}}{a^{2}} + C∫xsin(ax)dx=−axcos(ax)+a2sin(ax)+C(19)∫xcos(ax)dx=xsin(ax)a+cos(ax)a2+C\int x \cos(ax) dx = \frac{x \sin{\left(a x \right)}}{a} + \frac{\cos{\left(a x \right)}}{a^{2}} + C∫xcos(ax)dx=axsin(ax)+a2cos(ax)+C(20)∫x2sin(ax)dx=−x2cos(ax)a+2xsin(ax)a2+2cos(ax)a3+C\int x^2 \sin(ax) dx = - \frac{x^{2} \cos{\left(a x \right)}}{a} + \frac{2 x \sin{\left(a x \right)}}{a^{2}} + \frac{2 \cos{\left(a x \right)}}{a^{3}} + C∫x2sin(ax)dx=−ax2cos(ax)+a22xsin(ax)+a32cos(ax)+C(21)∫x2cos(ax)dx=x2sin(ax)a+2xcos(ax)a2−2sin(ax)a3+C\int x^2 \cos(ax) dx = \frac{x^{2} \sin{\left(a x \right)}}{a} + \frac{2 x \cos{\left(a x \right)}}{a^{2}} - \frac{2 \sin{\left(a x \right)}}{a^{3}} + C∫x2cos(ax)dx=ax2sin(ax)+a22xcos(ax)−a32sin(ax)+C(22)∫0∞tx−1e−tdt=Γ(x)\int_{0}^\infty t^{x-1}e^{-t} dt = \Gamma(x)∫0∞tx−1e−tdt=Γ(x)(23)∫xnsin(ax)dx=axn+2Γ(n2+1)1F2(n2+132,n2+2|−a2x24)2Γ(n2+2)\int x^n \sin(ax) dx = \frac{a x^{n + 2} \Gamma\left(\frac{n}{2} + 1\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{n}{2} + 1 \\ \frac{3}{2}, \frac{n}{2} + 2 \end{matrix}\middle| {- \frac{a^{2} x^{2}}{4}} \right)}}{2 \Gamma\left(\frac{n}{2} + 2\right)}∫xnsin(ax)dx=2Γ(2n+2)axn+2Γ(2n+1)1F2(2n+123,2n+2∣∣−4a2x2)(24)∫xncos(ax)dx=xn+1Γ(n2+12)1F2(n2+1212,n2+32|−a2x24)2Γ(n2+32)\int x^n \cos(ax) dx = \frac{x^{n + 1} \Gamma\left(\frac{n}{2} + \frac{1}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{n}{2} + \frac{1}{2} \\ \frac{1}{2}, \frac{n}{2} + \frac{3}{2} \end{matrix}\middle| {- \frac{a^{2} x^{2}}{4}} \right)}}{2 \Gamma\left(\frac{n}{2} + \frac{3}{2}\right)}∫xncos(ax)dx=2Γ(2n+23)xn+1Γ(2n+21)1F2(2n+2121,2n+23∣∣−4a2x2)(25)∫sin2(ax)dx=x2−sin(ax)cos(ax)2a+C,a≠0\int \sin^2(ax) dx = \frac{x}{2} - \frac{\sin{\left(a x \right)} \cos{\left(a x \right)}}{2a} + C, \quad a \neq 0∫sin2(ax)dx=2x−2asin(ax)cos(ax)+C,a=0(26)∫cos2(ax)dx=x2+sin(ax)cos(ax)2a+C,a≠0\int \cos^2(ax) dx = \frac{x}{2} + \frac{\sin{\left(a x \right)} \cos{\left(a x \right)}}{2a} + C, \quad a \neq 0∫cos2(ax)dx=2x+2asin(ax)cos(ax)+C,a=0(27)∫eax dx=1aeax+C\int e^{ax}~dx = \frac{1}{a} e^{ax} + C∫eax dx=a1eax+C(28)∫bax dx=baxalnb+C\int b^{ax}~ dx = \frac{b^{ax}}{a\ln b} + C∫bax dx=alnbbax+C(29)∫ln(ax+b)dx=baln∣ax+b∣−x+xln∣ax+b∣+C\int \ln(ax + b) dx = \frac{b}{a} \ln|ax + b| - x + x \ln|ax + b| + C∫ln(ax+b)dx=abln∣ax+b∣−x+xln∣ax+b∣+C(30)∫logc(ax+b)dx=xln(ax+b)ln(c)−xln(c)+C\int \log_{c}(ax + b) dx = \frac{x \ln{\left(a x + b \right)}}{\ln{\left(c \right)}} - \frac{x}{\ln{\left(c \right)}} + C∫logc(ax+b)dx=ln(c)xln(ax+b)−ln(c)x+C(31)∫dua2−u2=arcsin(ua)+C\int \frac{du}{\sqrt{a^2 - u^2}} = \operatorname{arcsin} \left(\frac{u}{a}\right) + C∫a2−u2du=arcsin(au)+C(32)∫dua2+u2=1aarctan(ua)+C\int \frac{du}{a^2 + u^2} = \frac{1}{a} \operatorname{arctan} \left(\frac{u}{a}\right) + C∫a2+u2du=a1arctan(au)+C(33)∫du∣u∣u2−a2=1aarcsec(ua)+C\int \frac{du}{|u|\sqrt{u^2 - a^2}} = \frac{1}{a} \operatorname{arcsec} \left(\frac{u}{a}\right) + C∫∣u∣u2−a2du=a1arcsec(au)+C(34)∫u′(x)u(x)dx=ln∣u(x)∣+C\int \frac{u'(x)}{u(x)}dx = \ln|u(x)| + C∫u(x)u′(x)dx=ln∣u(x)∣+C(35)∫−∞∞e−ax2dx=πa\int_{- \infty}^{\infty} e^{- a x^2} d x = \sqrt{\dfrac{\pi}{a}}∫−∞∞e−ax2dx=aπ(36)∫−∞∞xe−ax2dx=0\int_{-\infty}^\infty x e^{-ax^2} dx = 0∫−∞∞xe−ax2dx=0(37)∫−∞∞x2e−ax2dx=π4a3\int_{-\infty}^\infty x^2 e^{-ax^2}dx = \sqrt{ \frac{\pi}{4a^3} }∫−∞∞x2e−ax2dx=4a3π(38)∫xne−ax2 dx=(−1)na12−n2Γ(n2+12)2a+a12−n2Γ(n2+12)2a+C\int x^{n} e^{- a x^{2}}\, dx = \frac{\left(-1\right)^{n} a^{\frac{1}{2} - \frac{n}{2}} \Gamma\left(\frac{n}{2} + \frac{1}{2}\right)}{2 a} + \frac{a^{\frac{1}{2} - \frac{n}{2}} \Gamma\left(\frac{n}{2} + \frac{1}{2}\right)}{2 a} + C∫xne−ax2dx=2a(−1)na21−2nΓ(2n+21)+2aa21−2nΓ(2n+21)+C(39)∫abδ(x) dx=1,a<x<b\int_a^b \delta(x)\, dx = 1, \quad a < x < b∫abδ(x)dx=1,a<x<b(40)∫abf(x)δ(x−x′)dx=f(x′)\int_a^b f(x)\delta(x- x')dx = f(x')∫abf(x)δ(x−x′)dx=f(x′)(41)∫abf(x′)δ(x−x′)dx=f(x)\int_a^b f(x')\delta(x-x')dx = f(x)∫abf(x′)δ(x−x′)dx=f(x)(42)∫−∞∞sin(ax)bxdx=πb\int_{-\infty}^\infty \frac{\sin(ax)}{bx} dx = \frac{\pi}{b}∫−∞∞bxsin(ax)dx=bπ(43)