Integration Table

A reference table of common integrals (antiderivatives) used throughout physics and engineering.

GCE A-Level H2 Physics 2027

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This is a lookup table. In most physics problems, the hard part is getting the integral into one of these standard forms.

Simplify first (factor constants, expand/collect terms).
Try a substitution u = u(x) to match a known form.
Watch for “derivative in the numerator” patterns (e.g. ∫ (f'(x)/f(x))dx = ln |f(x)| + C).
Always differentiate your final answer once to check.

Table Of Basic Integrals

Basic Forms

∫ xⁿ dx = (1/(n + 1))xⁿ⁺¹, n ≠ -1

∫ (1/x)dx = ln ‖x‖

∫ u dv = uv - ∫ v du

∫ (1/(ax + b))dx = 1/a ln ‖ax + b‖

Integrals of Rational Functions

∫ (1/((x + a)²))dx = -1/(x + a)

∫ (x + a)ⁿ dx = ((x + a)ⁿ⁺¹)/(n + 1), n ≠ -1

∫ x(x + a)ⁿ dx = ((x + a)ⁿ⁺¹ ((n + 1)x-a))/((n + 1)(n + 2))

∫ (1/(1 + x²))dx = tan⁻¹ x

∫ (1/(a² + x²))dx = 1/a tan⁻¹ x/a

∫ (x/(a² + x²))dx = 1/2 ln ‖a² + x²‖

∫ (x²/(a² + x²))dx = x-a tan⁻¹ x/a

∫ (x³/(a² + x²))dx = (1/2)x²-(1/2)a² ln ‖a² + x²‖

∫ (1/(ax² + bx + c))dx = 2/(square root of (4ac-b²)) tan⁻¹ (2ax + b)/(square root of (4ac-b²))

∫ (1/((x + a)(x + b)))dx = 1/(b-a) ln(a + x)/(b + x), a ≠ b

∫ (x/((x + a)²))dx = a/(a + x) + ln ‖a + x‖

∫ (x/(ax² + bx + c))dx = 1/2a ln ‖ax² + bx + c‖ -b/(a square root of (4ac-b²)) tan⁻¹ (2ax + b)/(square root of (4ac-b²))

Integrals with Roots

∫ square root of (x-a) dx = (2/3)(x-a)^(3/2)

∫ 1/(square root of (x± a)) dx = 2 square root of (x± a)

∫ 1/(square root of (a-x)) dx = -2 square root of (a-x)

∫ x square root of (x-a) dx = { (2 a)/3 (x-a)^(3/2) + (2/5)(x-a)^(5/2), or; 2/3 x(x-a)^(3/2) - 4/15 (x-a)^(5/2), or; (2/15)(2a + 3x)(x-a)^(3/2) .

∫ square root of (ax + b) dx = (2b/3a + 2x/3) square root of (ax + b)

∫ (ax + b)^(3/2) dx = (2/5a)(ax + b)^(5/2)

∫ x/(square root of (x± a)) dx = (2/3)(x∓ 2a) square root of (x± a)

∫ square root of (x/(a-x)) dx = - square root of (x(a-x)) -a tan⁻¹ (square root of (x(a-x)))/(x-a)

∫ square root of (x/(a + x)) dx = square root of (x(a + x)) -a ln [square root of x + square root of (x + a)]

∫ x square root of (ax + b) dx = (2/(15 a²))(-2b² + abx + 3 a² x²) square root of (ax + b)

∫ square root of (x(ax + b)) dx = (1/(4a^(3/2)))[(2ax + b) square root of (ax(ax + b)) -b² ln ‖ a square root of x + square root of (a(ax + b)) ‖]

∫ square root of (x³(ax + b)) dx = [b/12a- b²/8a²x + x/3] square root of (x³(ax + b)) + b³/(8a^(5/2)) ln ‖ a square root of x + square root of (a(ax + b)) ‖

∫ square root of (x² ± a²) dx = (1/2)x square root of (x²± a²) ±(1/2)a² ln ‖ x + square root of (x²± a²) ‖

∫ square root of (a² - x²) dx = 1/2 x square root of (a²-x²) + (1/2)a² tan⁻¹ x/(square root of (a²-x²))

∫ x square root of (x² ± a²) dx = (1/3) (x² ± a²)^(3/2)

∫ 1/(square root of (x² ± a²)) dx = ln ‖ x + square root of (x² ± a²) ‖

∫ 1/(square root of (a² - x²)) dx = sin⁻¹ x/a

∫ x/(square root of (x²± a²)) dx = square root of (x² ± a²)

∫ x/(square root of (a²-x²)) dx = - square root of (a²-x²)

∫ x²/(square root of (x² ± a²)) dx = (1/2)x square root of (x² ± a²) ∓ (1/2)a² ln ‖ x + square root of (x²± a²) ‖

∫ square root of (a x² + b x + c) dx = ((b + 2ax)/4a) square root of (ax² + bx + c) + (4ac-b²)/(8a^(3/2)) ln ‖ 2ax + b + 2 square root of (a(ax² + bx⁺c)) ‖

∫ x square root of (a x² + bx + c) dx = (1/(48a^(5/2))) (2 square root of a square root of (ax² + bx + c) . (- 3b² + 2 abx + 8 a(c + ax²)); . + 3(b³-4abc) ln ‖b + 2ax + 2 square root of a square root of (ax² + bx + c) ‖)

∫1/(square root of (ax² + bx + c)) dx = 1/(square root of a) ln ‖ 2ax + b + 2 square root of (a(ax² + bx + c)) ‖

∫ x/(square root of (ax² + bx + c)) dx = (1/a) square root of (ax² + bx + c) - b/(2a^(3/2)) ln ‖ 2ax + b + 2 square root of (a(ax² + bx + c)) ‖

∫dx/((a² + x²)^(3/2)) = x/(a² square root of (a² + x²))

Integrals with Logarithms

∫ ln ax dx = x ln ax - x

∫ x ln x dx = 1/2 x² ln x-x²/4

∫ x² ln x dx = 1/3 x³ ln x-x³/9

∫ xⁿ ln x dx = xⁿ⁺¹((ln x)/(n + 1)-1/((n + 1)²)), n ≠ -1

∫ (ln ax)/x dx = (1/2) (ln ax)²

∫ (ln x)/x² dx = -1/x-(ln x)/x

∫ ln(ax + b) dx = (x + b/a) ln(ax + b) - x, a ≠ 0

∫ ln(x² + a²) dx = x ln(x² + a²) + 2a tan⁻¹ x/a - 2x

∫ ln(x² - a²) dx = x ln(x² - a²) + a ln(x + a)/(x-a) - 2x

∫ ln(ax² + bx + c) dx = (1/a) square root of (4ac-b²) tan⁻¹ (2ax + b)/(square root of (4ac-b²)) -2x + (b/2a + x) ln(ax² + bx + c)

∫ x ln(ax + b) dx = bx/2a-(1/4)x² + (1/2)(x²-b²/a²) ln(ax + b)

∫ x ln(a² - b² x²) dx = -(1/2)x² + (1/2)(x² - a²/b²) ln(a² -b² x²)

∫ (ln x)² dx = 2x - 2x ln x + x (ln x)²

∫ (ln x)³ dx = -6 x + x (ln x)³-3 x (ln x)² + 6 x ln x

∫ x (ln x)² dx = x²/4 + 1/2 x² (ln x)²-1/2 x² ln x

∫ x² (ln x)² dx = (2 x³)/27 + 1/3 x³ (ln x)²-2/9 x³ ln x

Integrals with Exponentials

∫ e^ax dx = (1/a)e^ax

∫ square root of x e^ax dx = (1/a) square root of x e^ax + (i square root of π)/(2a^(3/2)) erf(i square root of ax), where erf(x) = 2/(square root of π)∫₀^x e^(-t²)dt

∫ x e^x dx = (x-1) e^x

∫ x e^ax dx = (x/a-1/a²) e^ax

∫ x² e^x dx = (x² - 2x + 2) e^x

∫ x² e^ax dx = (x²/a-2x/a² + 2/a³) e^ax

∫ x³ e^x dx = (x³-3x² + 6x - 6) e^x

∫ xⁿ e^ax dx = (xⁿ e^ax)/a - n/a∫ xⁿ⁻¹e^ax dx

∫ xⁿ e^ax dx = (((-1)ⁿ)/aⁿ⁺¹)Γ(1 + n,-ax), where Γ(a,x) = ∫ₓ^∞ t^(a-1)e^(-t) dt

∫ e^ax² dx = -((i square root of π)/(2 square root of a))erf(ix square root of a)

∫ e^(-ax²) dx = ((square root of π)/(2 square root of a))erf(x square root of a)

∫ x e^(-ax²) dx = -(1/2a)e^(-ax²)

∫ x² e^(-ax²) dx = (1/4) square root of (π/a³) erf(x square root of a) -(x/2a)e^(-ax²)

Integrals with Trigonometric Functions

∫ sin ax dx = -1/a cos ax

∫ sin² ax dx = x/2 - (sin 2ax)/4a

∫ sin³ ax dx = -(3 cos ax)/4a + (cos 3ax)/12a

∫ sinⁿ ax dx = -(1/a) cos ax ₂F₁[1/2, (1-n)/2, 3/2, cos² ax]

∫ cos ax dx = 1/a sin ax

∫ cos² ax dx = x/2 + (sin 2ax)/4a

∫ cos³ ax dx = (3 sin ax)/4a + (sin 3ax)/12a

∫ cos ^p ax dx = -(1/(a(1 + p))) cos ^(1 + p) ax × ₂F₁[(1 + p)/2, 1/2, (3 + p)/2, cos² ax]

∫ cos x sin x dx = 1/2 sin² x + c₁ = -1/2 cos² x + c₂ = -1/4 cos 2x + c₃

∫ cos ax sin bx dx = (cos((a-b)x))/(2(a-b)) - (cos((a + b)x))/(2(a + b)), a ≠ b

∫ sin² ax cos bx dx = -(sin((2a-b)x))/(4(2a-b)) + (sin bx)/2b - (sin((2a + b)x))/(4(2a + b))

∫ sin² x cos x dx = 1/3 sin³ x

∫ cos² ax sin bx dx = (cos((2a-b)x))/(4(2a-b)) - (cos bx)/2b - (cos((2a + b)x))/(4(2a + b))

∫ cos² ax sin ax dx = -1/3a cos³ ax

∫ sin² ax cos² bx dx = x/4 -(sin 2ax)/8a- (sin(2(a-b)x))/(16(a-b)) + (sin 2bx)/8b- (sin(2(a + b)x))/(16(a + b))

∫ sin² ax cos² ax dx = x/8-(sin 4ax)/32a

∫ tan ax dx = -1/a ln cos ax

∫ tan² ax dx = -x + 1/a tan ax

∫ tanⁿ ax dx = (tanⁿ⁺¹ ax)/(a(1 + n)) × ₂F₁((n + 1)/2, 1, (n + 3)/2, - tan² ax)

∫ tan³ ax dx = 1/a ln cos ax + 1/2a sec² ax

∫ sec x dx = ln ‖ sec x + tan x ‖ = 2 tanh⁻¹ (tan x/2)

∫ sec² ax dx = 1/a tan ax

∫ sec³ x dx = 1/2 sec x tan x + 1/2 ln ‖ sec x + tan x ‖

∫ sec x tan x dx = sec x

∫ sec² x tan x dx = 1/2 sec² x

∫ secⁿ x tan x dx = 1/n secⁿ x, n ≠ 0

∫ csc x dx = ln ‖ tan x/2 ‖ = ln ‖ csc x - cot x‖ + C

∫ csc² ax dx = -1/a cot ax

∫ csc³ x dx = -1/2 cot x csc x + 1/2 ln ‖ csc x - cot x ‖

∫ cscⁿ x cot x dx = -1/n cscⁿ x, n ≠ 0

∫ sec x csc x dx = ln ‖ tan x ‖

Products of Trigonometric Functions and Monomials

∫ x cos x dx = cos x + x sin x

∫ x cos ax dx = 1/a² cos ax + x/a sin ax

∫ x² cos x dx = 2 x cos x + (x² - 2) sin x

∫ x² cos ax dx = (2 x cos ax)/a² + (a² x² - 2)/a³ sin ax

∫ xⁿ cos x dx = -(1/2)(i)ⁿ⁺¹[Γ(n + 1, -ix) + (-1)ⁿ Γ(n + 1, ix)]

∫ xⁿ cos ax dx = (1/2)(ia)¹⁻ⁿ[(-1)ⁿ Γ(n + 1, -iax) -Γ(n + 1, ixa)]

∫ x sin x dx = -x cos x + sin x

∫ x sin ax dx = -(x cos ax)/a + (sin ax)/a²

∫ x² sin x dx = (2-x²) cos x + 2 x sin x

∫ x² sin ax dx = (2-a²x²)/a³ cos ax + (2 x sin ax)/a²

∫ xⁿ sin x dx = -(1/2)(i)ⁿ[Γ(n + 1, -ix) - (-1)ⁿΓ(n + 1, -ix)]

∫ x cos² x dx = x²/4 + 1/8 cos 2x + 1/4 x sin 2x

∫ x sin² x dx = x²/4-1/8 cos 2x - 1/4 x sin 2x

∫ x tan² x dx = -x²/2 + ln cos x + x tan x

∫ x sec² x dx = ln cos x + x tan x

Products of Trigonometric Functions and Exponentials

∫ e^x sin x dx = (1/2)e^x (sin x - cos x)

∫ e^bx sin ax dx = (1/(a² + b²))e^bx (b sin ax - a cos ax)

∫ e^x cos x dx = (1/2)e^x (sin x + cos x)

∫ e^bx cos ax dx = 1/(a² + b²) e^bx (a sin ax + b cos ax)

∫ x e^x sin x dx = (1/2)e^x (cos x - x cos x + x sin x)

∫ x e^x cos x dx = (1/2)e^x (x cos x - sin x + x sin x)

Integrals of Hyperbolic Functions

∫ cosh ax dx = 1/a sinh ax

∫ e^ax cosh bx dx = { e^ax/(a²-b²) (a cosh bx - b sinh bx) a ≠ b; e^2ax/4a + x/2 a = b

∫ sinh ax dx = 1/a cosh ax

∫ e^ax sinh bx dx = { e^ax/(a²-b²) (-b cosh bx + a sinh bx) a ≠ b; e^2ax/4a - x/2 a = b

∫ tanh ax dx = 1/a ln cosh ax

∫ e^ax tanh bx dx = { (e^((a + 2b)x))/(a + 2b) ₂F₁[1 + a/2b,1,2 + a/2b, -e^2bx]; -(1/a)e^ax₂F₁[1, a/2b,1 + a/2b, -e^2bx] a ≠ b; (e^ax-2 tan⁻¹ (e^ax))/a a = b

∫ cos ax cosh bx dx = 1/(a² + b²) [a sin ax cosh bx + b cos ax sinh bx]

∫ cos ax sinh bx dx = 1/(a² + b²) [b cos ax cosh bx + a sin ax sinh bx]

∫ sin ax cosh bx dx = 1/(a² + b²) [-a cos ax cosh bx + b sin ax sinh bx]

∫ sin ax sinh bx dx = 1/(a² + b²) [b cosh bx sin ax - a cos ax sinh bx]

∫ sinh ax cosh ax dx = (1/4a)[-2ax + sinh 2ax]

∫ sinh ax cosh bx dx = (1/(b²-a²))[b cosh bx sinh ax - a cosh ax sinh bx]

Useful Integral Results For Mechanics

∫^(+∞)_(- ∞) e^(-ax²) = square root of (π/a) ∫^(+∞)_(- ∞)x²ⁿ e^(-ax²) = (-1)ⁿ (∂ⁿ/(∂ aⁿ)) square root of (π/a) ∫^(+∞)_(- ∞) e^(-ax² + bx) = e^(b²/4a) square root of (π/a) ∫^(+a/2)_(-a/2) x² sin² ((n π x)/a) = 1/24 a³ (1 - (6(-1)ⁿ)/(n² π²)) ∫^(+a/2)_(-a/2) x² cos² ((n π x)/a) = 1/24 a³ (1 + (6(-1)ⁿ)/(n² π²)) ∫^(+a/2)_(-a/2) x cos((π x)/a) sin((2 π x)/a) = 8a²/(9 π ²) _ _ _ _ _ _ _ _ _ _ _ ∫^a_b dx/(square root of ((a-x) (x-b))) = π for a > b ∫^a_b dx/(x square root of ((a-x) (x-b))) = π/(square root of ab) for a > b > 0 ∫^(π/2)_(- π/2) dx/(1 + y sin x) = π/(square root of (1 - y²)) for -1 < y < 1

Useful Integrals For Electromagnetism:

∫ dx/(square root of (a² - x²)) = arcsin x/a ∫ (x dx)/(square root of (a² + x²)) = square root of (a² + x²) ∫ dx/(square root of (a² + x²)) = ln(x + square root of (a² + x²)) ∫ dx/(a² + x²) = 1/a arctan x/a ∫ dx/((a² + x²)^(3/2)) = 1/a² x/(square root of (a² + x²)) ∫(x dx)/((a² + x²)^(3/2)) = - 1/(square root of (a² + x²)) ∫ dx/(square root of ((x - a)² + b²)) = ln 1/((a - x) + square root of ((a-x)² + b²)) ∫ ((x - a) dx)/([(x-a)² + b²]^(3/2)) = - 1/(square root of ((x-a)² + b²)) ∫ dx/([(x - a)² + b²]^(3/2)) = (x - a)/(b² square root of ((x - a)² + b²)) Back To Useful Mathematics References

Quick worked examples

A. ∫ 2x/(x² + 1) dx

Let u = x² + 1, so du = 2x dx:

∫ 2x/(x² + 1) dx = ∫ 1/u du = ln |u| + C = ln(x² + 1) + C.

B. ∫ e^3x cos(2x) dx

This matches the common “e^ax cos(bx)” pattern. One result is:

∫ e^ax cos(bx) dx = (e^ax/(a² + b²))(a cos bx + b sin bx) + C.

With a = 3 and b = 2:

∫ e^3x cos(2x) dx = (e^3x/13)(3 cos 2x + 2 sin 2x) + C.

Back To Useful Mathematics References