Table of Useful Integrals

Q: What types of integral formulas are included in the table?

The document provides multiple integral identities, including definite/indefinite forms and expressions involving exponential, sine, and cosine terms. It also lists results related to products of trigonometric functions and powers.

Q: Does the table include series expansions?

Yes. It contains formulas for Taylor Series and Geometric Series in summation form.

Q: What other mathematical topics appear besides integrals?

It includes Euler’s formula, solutions related to quadratic and higher-order polynomials, a constant-coefficient second-order homogeneous differential equation form, spherical polar to Cartesian coordinate conversions, and commutator/operator identities. It also mentions atomic units and physical constants as sections.

Oct 14, 2026

Upload by Riley West

A reference collection of mathematical results centered on useful integrals and related analytic formulas. The content includes definite and indefinite integration identities, common transformations such as integration by parts, and standard series expansions including Taylor and geometric series. It also provides selected advanced formulae for Euler’s formula, quadratic and polynomial solutions, differential equation solutions with constant coefficients, coordinate conversions in spherical polar coordinates, commutator identities, and creation/annihilation operator relations, plus atomic units placeholders.

LanguageEnglish

CategoryGeneral

FormatPDF

Pages6

Reading Time2 min

AccessFree to download

Useful Integrals

Integration identities and by-parts results

Series expansions and core formulas

Differential equations and coordinate conversions

Commutator and operator identities

Table of Useful Integrals, etc. ∞ 2 1 􀀣 π 􀀦 e− ax dx = 2 􀀥􀀤 ~~ ~~a~~ ~~ 􀀧􀀨 ∞ 2 − ax2 1 􀀣 x e dx = 4a 􀀥􀀤 12 π 􀀦 a 􀀧􀀨 12 x 2ne − ax2 dx = 1⋅~~ ~~3~~ ~~⋅~~ ~~52⋅n⋅+⋅1nn~~ ~~−~~ ~~1~~) ~~0 ∞ n! ∫x ne− ax dx = an+1 0 􀀤 π 􀀧􀀦􀀥 a 􀀨􀀩 12 ∞∫ 0 ∞∫ 0 ∞∫ 0 2 1 xe − ax dx = 2a x 3e − ax2 dx = 21a2 x 2n+1e − ax2 dx = 2nan!+1 Integration by Parts: b b ∫UdV = UVba − ∫VdU U and V are functions of x. Integrate from x = a to x = b a a ∫sin (ax ) dx = − ~~ ~~1a cos( ax ) ∫sin2 ( ax ) dx = x2 − sin4~~( ~~2aax~~ )~~ ∫sin3 ( ax ) dx = − ~~ ~~1a cos( ax ) + 31a cos3 ( ax ) ∫sin4 ( ax )dx = 3x8 − 3sin16~~( ~~2aax~~ )~~ − sin3~~ (~~ax4~~)~~caos~~ ( ~~ax~~ )~~ ∫sin (ax )sin (bx ) dx = sin~~(~~(~~ ~~aa~~ ~~−~~ ~~bb~~)~~)~~ ~~x~~ ~~~~ ~~ − sin~~(~~(~~ ~~aa~~ ~~~~ ~~bb~~)~~)~~ ~~x~~ ~~~~ ~~ where a2 ≠ b2 ∫cos( ax )cos( bx ) dx = sin~~(~~(~~ ~~aa~~ ~~−~~ ~~bb~~)~~)~~ ~~x~~ ~~~~ ~~ + sin~~(~~(~~ ~~aa~~ ~~~~ ~~bb~~)~~)~~ ~~x~~ ~~ ∫sin (ax )cos( bx ) dx = −~~ ~~cos2􀀢~~􀀣(~~a(~~ ~~a−−bb~~) ~~)x~~ ~~~~ ~~ − co~~(~~(aa~~ ~~~~ ~~bb~~)~~)~~ ~~x~~ ~~ ∫x sin2 ( ax ) dx = x42~~ ~~ − x~~ ~~sin4~~(~~2aax~~ )~~ − cos8~~(~~2a2ax~~ )~~ ∫x2 sin2 ( ax ) dx = x63 − 􀀢􀀣􀀤 x24a − 81a3~~ ~~ 􀀥􀀦􀀧 sin (2ax) − x~~ ~~cos4a~~( ~~22ax~~ )~~ ∫x sin (ax )sin (bx ) dx = cos2(−−bb)x~~ ~~~~ ~~ − cos2(++bb)x~~ ~~~~ ~~ + x~~ ~~sin2(a(~~ ~~b)2)x~~ ~~~~ ~~ − x~~ ~~sin2(a(~~ ~~b)2)x~~ ~~ ∫x sin (ax ) dx = sina~~(~~ax2~~)~~ − x~~ ~~cosa~~( ~~ax~~ )~~ ∫x cos( ax ) dx = x~~ ~~sina~~( ~~ax~~ )~~ + cosa~~(~~2ax~~ )~~ ∫cos( ax ) dx = sina~~( ~~ax~~ )~~ ∫cos2 ( ax ) dx = x2 + sin4~~( ~~2aax~~ )~~ ∫x2 cos2 ( ax ) dx = x63 + 􀀢􀀣􀀤 x24a − 81a3~~ ~~ 􀀥􀀦􀀧 sin (2ax) + x~~ ~~cos4a~~( ~~22ax~~ )~~ ∫cos( bx )e − ax2 dx = ~~(~~a2eb2~~ )~~ acos( bx ) + bsin (bx ) Taylor Series: ∑ nn∞0 ~~ ~~f~~ ~~(nn)~~ (~~!xo~~ )~~ (x − xo )n Geometric Series: ∞ 1 0 xn = 1~~ ~~−~~ ~~x | Euler’s Formula:
eiφ = cosφ + isinφ | | --- | | Quadratic Equation and other higher order polynomials:
ax 2 + bx + c = 0
−b ± b2 − 4ac x =
2a
ax 4 + bx2 + c = 0
􀀢 −b ± b2 − 4ac 􀀥 x = ± 􀀣􀀤 ~~ ~~2a~~ ~~ 􀀦􀀧 | | General Solution for a Second Order Homogeneous Differential Equation with Constant Coefficients:
If: y ʹ′ʹ′ + py ʹ′ + qy = 0
Assume a solution for y:
y = esx y ʹ′ = sesx y ʹ′ʹ′ = s2 esx
∴ s2 esx + psesx + qesx = 0 and s2 + ps + q = 0 Hence y = c1es1x + c2 es2x | | Conversions from spherical polar coordinates into Cartesian coordinates:
x = r sinθ cosφ y = r sinθ sinφ x = r cosθ
dv = r2 sinθdrdθdφ 0 < r < ∞
0 < θ < π
0 < φ < 2π | Commutator Identities: ˆA, ˆB = − ˆB, ˆA ˆA, ˆAn = 0 n = 1, 2, 3, 􀀡 kˆA, ˆB = ˆA, kˆB = k ˆA, ˆB ˆA + ˆB, ˆC = ˆA, ˆC + ˆB, ˆC ˆA, ˆBˆC = ˆA, ˆB ˆC + ˆB ˆA, ˆC ˆAˆB, ˆC = ˆA, ˆC ˆB + ˆA ˆB, ˆC Creation and Annihilation Operators l± l , ml , s , ms = ( l ( l + 1) − ml ( ml ± 1))2 􀀡 l , ml ± 1 , s , ms s± l , ml , s , ms = (s (s + 1) − ms ( ms ± 1))2 􀀡 l , ml , s , ms ±1 j± j , mj = ( j ( j + 1) − mj ( mj ± 1))2 􀀡 j , mj ±1 Atomic Units: Physical Constants: Operators:

FAQs

What types of integral formulas are included in the table?

The document provides multiple integral identities, including definite/indefinite forms and expressions involving exponential, sine, and cosine terms. It also lists results related to products of trigonometric functions and powers.

Does the table include series expansions?

Yes. It contains formulas for Taylor Series and Geometric Series in summation form.

What other mathematical topics appear besides integrals?

It includes Euler’s formula, solutions related to quadratic and higher-order polynomials, a constant-coefficient second-order homogeneous differential equation form, spherical polar to Cartesian coordinate conversions, and commutator/operator identities. It also mentions atomic units and physical constants as sections.

Oct 14, 2026

Upload by Riley West