Calculus II — Integration & Trigonometry Formula Compendium

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\n PART I\n

Integration

\n indefinite \\( +\\,C \\) omitted-implied · definite at end\n

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Fundamental rules of integration

linearity & the power rule

Constant\\[ \\int k\\,dx = kx + C \\]

Constant multiple\\[ \\int k\\,f(x)\\,dx = k\\!\\int f(x)\\,dx \\]

Sum / difference\\[ \\int\\!\\big[f\\pm g\\big]dx = \\int\\! f\\,dx \\pm \\int\\! g\\,dx \\]

Power rule\\[ \\int x^{n}\\,dx = \\frac{x^{n+1}}{n+1}+C \\]n ≠ −1

Reciprocal\\[ \\int \\frac{1}{x}\\,dx = \\ln|x| + C \\]

Linear inner term\\[ \\int (ax+b)^{n}dx = \\frac{(ax+b)^{n+1}}{a(n+1)}+C \\]n ≠ −1

Logarithmic form\\[ \\int \\frac{f'(x)}{f(x)}\\,dx = \\ln|f(x)| + C \\]

Power × derivative\\[ \\int [f(x)]^{n}f'(x)\\,dx = \\frac{[f(x)]^{n+1}}{n+1}+C \\]

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Roots, reciprocals & logarithms

common rewrites

Square root\\[ \\int \\sqrt{x}\\,dx = \\tfrac{2}{3}x^{3/2}+C \\]

Reciprocal square\\[ \\int \\frac{1}{x^{2}}\\,dx = -\\frac{1}{x}+C \\]

Natural log\\[ \\int \\ln x\\,dx = x\\ln x - x + C \\]

Log base a\\[ \\int \\log_a x\\,dx = \\frac{x\\ln x - x}{\\ln a}+C \\]

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Exponential functions

Natural base\\[ \\int e^{x}\\,dx = e^{x}+C \\]

Scaled exponent\\[ \\int e^{ax}\\,dx = \\frac{1}{a}e^{ax}+C \\]

General base\\[ \\int a^{x}\\,dx = \\frac{a^{x}}{\\ln a}+C \\]a > 0, a ≠ 1

x · exponential\\[ \\int x\\,e^{x}\\,dx = (x-1)e^{x}+C \\]

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Trigonometric integrals

the six functions & their squares

Sine\\[ \\int \\sin x\\,dx = -\\cos x + C \\]

Cosine\\[ \\int \\cos x\\,dx = \\sin x + C \\]

Tangent\\[ \\int \\tan x\\,dx = \\ln|\\sec x| + C \\]

Cotangent\\[ \\int \\cot x\\,dx = \\ln|\\sin x| + C \\]

Secant\\[ \\int \\sec x\\,dx = \\ln|\\sec x + \\tan x| + C \\]

Cosecant\\[ \\int \\csc x\\,dx = \\ln|\\csc x - \\cot x| + C \\]

Secant squared\\[ \\int \\sec^{2}x\\,dx = \\tan x + C \\]

Cosecant squared\\[ \\int \\csc^{2}x\\,dx = -\\cot x + C \\]

sec · tan\\[ \\int \\sec x\\tan x\\,dx = \\sec x + C \\]

csc · cot\\[ \\int \\csc x\\cot x\\,dx = -\\csc x + C \\]

Sine squared\\[ \\int \\sin^{2}x\\,dx = \\frac{x}{2}-\\frac{\\sin 2x}{4}+C \\]

Cosine squared\\[ \\int \\cos^{2}x\\,dx = \\frac{x}{2}+\\frac{\\sin 2x}{4}+C \\]

\n Powers of sine & cosine. If a power is odd, peel one factor off and convert the rest with \\(\\sin^2+\\cos^2=1\\), then substitute. If both are even, knock the powers down with the power-reducing identities (Part II §9) until they integrate directly.\n

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Integrals giving inverse-trig & log forms

memorize the three “arc” patterns

→ arcsin\\[ \\int \\frac{dx}{\\sqrt{a^{2}-x^{2}}} = \\arcsin\\frac{x}{a}+C \\]

→ arctan\\[ \\int \\frac{dx}{a^{2}+x^{2}} = \\frac{1}{a}\\arctan\\frac{x}{a}+C \\]

→ arcsec\\[ \\int \\frac{dx}{x\\sqrt{x^{2}-a^{2}}} = \\frac{1}{a}\\,\\operatorname{arcsec}\\Big|\\frac{x}{a}\\Big|+C \\]

→ log (sum)\\[ \\int \\frac{dx}{\\sqrt{x^{2}+a^{2}}} = \\ln\\!\\big|x+\\sqrt{x^{2}+a^{2}}\\big|+C \\]

→ log (diff)\\[ \\int \\frac{dx}{\\sqrt{x^{2}-a^{2}}} = \\ln\\!\\big|x+\\sqrt{x^{2}-a^{2}}\\big|+C \\]

Partial-fraction form\\[ \\int \\frac{dx}{a^{2}-x^{2}} = \\frac{1}{2a}\\ln\\!\\Big|\\frac{a+x}{a-x}\\Big|+C \\]

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Hyperbolic functions

sinh\\[ \\int \\sinh x\\,dx = \\cosh x + C \\]

cosh\\[ \\int \\cosh x\\,dx = \\sinh x + C \\]

tanh\\[ \\int \\tanh x\\,dx = \\ln(\\cosh x) + C \\]

coth\\[ \\int \\coth x\\,dx = \\ln|\\sinh x| + C \\]

sech²\\[ \\int \\operatorname{sech}^{2}x\\,dx = \\tanh x + C \\]

csch²\\[ \\int \\operatorname{csch}^{2}x\\,dx = -\\coth x + C \\]

sech · tanh\\[ \\int \\operatorname{sech}x\\tanh x\\,dx = -\\operatorname{sech}x + C \\]

csch · coth\\[ \\int \\operatorname{csch}x\\coth x\\,dx = -\\operatorname{csch}x + C \\]

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Methods of integration

when no rule fits directly

Substitution (u-sub)\\[ \\int f\\big(g(x)\\big)\\,g'(x)\\,dx = \\int f(u)\\,du,\\qquad u=g(x),\\; du=g'(x)\\,dx \\]

Integration by parts\\[ \\int u\\,dv = uv - \\int v\\,du \\]choose u by LIATE: Log · Inverse · Algebraic · Trig · Exponential

Trig sub · type √(a²−x²)\\[ x = a\\sin\\theta \\]

Trig sub · type √(a²+x²)\\[ x = a\\tan\\theta \\]

Trig sub · type √(x²−a²)\\[ x = a\\sec\\theta \\]

Weierstrass (half-angle)\\[ t=\\tan\\tfrac{x}{2}:\\;\\; \\sin x=\\tfrac{2t}{1+t^{2}},\\;\\cos x=\\tfrac{1-t^{2}}{1+t^{2}} \\]

Reduction — sine\\[ \\int \\sin^{n}x\\,dx = -\\frac{\\sin^{n-1}x\\cos x}{n} + \\frac{n-1}{n}\\int \\sin^{n-2}x\\,dx \\]

Reduction — cosine\\[ \\int \\cos^{n}x\\,dx = \\frac{\\cos^{n-1}x\\sin x}{n} + \\frac{n-1}{n}\\int \\cos^{n-2}x\\,dx \\]

\n Partial fractions. For a proper rational function \\( \\tfrac{P(x)}{Q(x)} \\), factor \\(Q\\) completely, then split into a sum: each linear factor \\((ax+b)\\) contributes \\(\\tfrac{A}{ax+b}\\); a repeated factor \\((ax+b)^k\\) contributes one term per power up to \\(k\\); each irreducible quadratic \\((ax^2+bx+c)\\) contributes \\(\\tfrac{Bx+C}{ax^2+bx+c}\\). If the fraction is improper (degree of \\(P \\ge\\) degree of \\(Q\\)), do polynomial long division first.\n

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Definite integrals & the Fundamental Theorem

FTC · Part 1 (evaluation)\\[ \\int_{a}^{b} f(x)\\,dx = F(b)-F(a),\\qquad F'=f \\]

FTC · Part 2 (derivative of an integral)\\[ \\frac{d}{dx}\\int_{a}^{x} f(t)\\,dt = f(x) \\]

Empty interval\\[ \\int_{a}^{a} f\\,dx = 0 \\]

Reverse limits\\[ \\int_{a}^{b} f\\,dx = -\\int_{b}^{a} f\\,dx \\]

Split (additivity)\\[ \\int_{a}^{b}\\! f = \\int_{a}^{c}\\! f + \\int_{c}^{b}\\! f \\]

Even function\\[ \\int_{-a}^{a} f = 2\\int_{0}^{a} f \\]if f(−x)=f(x)

Odd function\\[ \\int_{-a}^{a} f = 0 \\]if f(−x)=−f(x)

Average value\\[ \\bar f = \\frac{1}{b-a}\\int_{a}^{b} f\\,dx \\]

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\n PART II\n

Trigonometry

\n identities valid for all admissible angles\n

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Definitions, reciprocals & quotients

SOH·CAH·TOA

Sine\\[ \\sin\\theta = \\frac{\\text{opp}}{\\text{hyp}} \\]

Cosine\\[ \\cos\\theta = \\frac{\\text{adj}}{\\text{hyp}} \\]

Tangent\\[ \\tan\\theta = \\frac{\\text{opp}}{\\text{adj}} \\]

Reciprocal — csc\\[ \\csc\\theta = \\frac{1}{\\sin\\theta} \\]

Reciprocal — sec\\[ \\sec\\theta = \\frac{1}{\\cos\\theta} \\]

Reciprocal — cot\\[ \\cot\\theta = \\frac{1}{\\tan\\theta} \\]

Quotient — tan\\[ \\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta} \\]

Quotient — cot\\[ \\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta} \\]

\n Radian ↔ degree. \\( \\pi \\text{ rad} = 180^{\\circ} \\), so \\(1\\text{ rad}=\\tfrac{180}{\\pi}^{\\!\\circ}\\) and \\(1^{\\circ}=\\tfrac{\\pi}{180}\\text{ rad}\\). On the unit circle a point is \\((\\cos\\theta,\\ \\sin\\theta)\\); signs by quadrant follow ASTC — All, Sine, Tangent, Cosine positive in QI–QIV.\n

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Pythagorean · even–odd · cofunction

Pythagorean — base\\[ \\sin^{2}\\theta + \\cos^{2}\\theta = 1 \\]

Pythagorean — tan\\[ 1 + \\tan^{2}\\theta = \\sec^{2}\\theta \\]

Pythagorean — cot\\[ 1 + \\cot^{2}\\theta = \\csc^{2}\\theta \\]

Even — cosine\\[ \\cos(-\\theta) = \\cos\\theta \\]

Odd — sine\\[ \\sin(-\\theta) = -\\sin\\theta \\]

Odd — tangent\\[ \\tan(-\\theta) = -\\tan\\theta \\]

Cofunction — sin/cos\\[ \\sin\\!\\big(\\tfrac{\\pi}{2}-\\theta\\big) = \\cos\\theta \\]

Cofunction — tan/cot\\[ \\tan\\!\\big(\\tfrac{\\pi}{2}-\\theta\\big) = \\cot\\theta \\]

Cofunction — sec/csc\\[ \\sec\\!\\big(\\tfrac{\\pi}{2}-\\theta\\big) = \\csc\\theta \\]

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Sum & difference formulas

Sine\\[ \\sin(A\\pm B) = \\sin A\\cos B \\pm \\cos A\\sin B \\]

Cosine\\[ \\cos(A\\pm B) = \\cos A\\cos B \\mp \\sin A\\sin B \\]

Tangent\\[ \\tan(A\\pm B) = \\frac{\\tan A \\pm \\tan B}{1 \\mp \\tan A\\tan B} \\]

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Double-angle · half-angle · power-reducing

Double — sine\\[ \\sin 2\\theta = 2\\sin\\theta\\cos\\theta \\]

Double — cosine\\[ \\cos 2\\theta = \\cos^{2}\\theta-\\sin^{2}\\theta \\]

Double — cosine (alt)\\[ \\cos 2\\theta = 2\\cos^{2}\\theta-1 = 1-2\\sin^{2}\\theta \\]

Double — tangent\\[ \\tan 2\\theta = \\frac{2\\tan\\theta}{1-\\tan^{2}\\theta} \\]

Half — sine\\[ \\sin\\frac{\\theta}{2} = \\pm\\sqrt{\\frac{1-\\cos\\theta}{2}} \\]

Half — cosine\\[ \\cos\\frac{\\theta}{2} = \\pm\\sqrt{\\frac{1+\\cos\\theta}{2}} \\]

Half — tangent\\[ \\tan\\frac{\\theta}{2} = \\frac{1-\\cos\\theta}{\\sin\\theta} = \\frac{\\sin\\theta}{1+\\cos\\theta} \\]

Power-reducing — sin²\\[ \\sin^{2}\\theta = \\frac{1-\\cos 2\\theta}{2} \\]

Power-reducing — cos²\\[ \\cos^{2}\\theta = \\frac{1+\\cos 2\\theta}{2} \\]

Power-reducing — tan²\\[ \\tan^{2}\\theta = \\frac{1-\\cos 2\\theta}{1+\\cos 2\\theta} \\]

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Product-to-sum & sum-to-product

Product → sum · sin·cos\\[ \\sin A\\cos B = \\tfrac{1}{2}\\big[\\sin(A+B)+\\sin(A-B)\\big] \\]

Product → sum · cos·cos\\[ \\cos A\\cos B = \\tfrac{1}{2}\\big[\\cos(A-B)+\\cos(A+B)\\big] \\]

Product → sum · sin·sin\\[ \\sin A\\sin B = \\tfrac{1}{2}\\big[\\cos(A-B)-\\cos(A+B)\\big] \\]

Sum → product · sines\\[ \\sin A + \\sin B = 2\\sin\\tfrac{A+B}{2}\\cos\\tfrac{A-B}{2} \\]

Diff → product · sines\\[ \\sin A - \\sin B = 2\\cos\\tfrac{A+B}{2}\\sin\\tfrac{A-B}{2} \\]

Sum → product · cosines\\[ \\cos A + \\cos B = 2\\cos\\tfrac{A+B}{2}\\cos\\tfrac{A-B}{2} \\]

Diff → product · cosines\\[ \\cos A - \\cos B = -2\\sin\\tfrac{A+B}{2}\\sin\\tfrac{A-B}{2} \\]

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Laws for any triangle

sides a,b,c opposite angles A,B,C

Law of sines\\[ \\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C} = 2R \\]

Law of cosines\\[ c^{2} = a^{2}+b^{2}-2ab\\cos C \\]

Law of tangents\\[ \\frac{a-b}{a+b} = \\frac{\\tan\\frac{A-B}{2}}{\\tan\\frac{A+B}{2}} \\]

Area — two sides & angle\\[ \\text{Area} = \\tfrac{1}{2}ab\\sin C \\]

Area — Heron's formula\\[ \\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)},\\qquad s=\\tfrac{a+b+c}{2} \\]

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Special-angle values & inverse functions

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“und.” = undefined. Multiply patterns: \\(\\sin\\) runs \\(\\tfrac{\\sqrt0}{2},\\tfrac{\\sqrt1}{2},\\dots,\\tfrac{\\sqrt4}{2}\\) across the row.
\\(\\theta\\) (deg)	0°	30°	45°	60°	90°
\\(\\theta\\) (rad)	\\(0\\)	\\(\\frac{\\pi}{6}\\)	\\(\\frac{\\pi}{4}\\)	\\(\\frac{\\pi}{3}\\)	\\(\\frac{\\pi}{2}\\)
\\(\\sin\\theta\\)	\\(0\\)	\\(\\frac{1}{2}\\)	\\(\\frac{\\sqrt2}{2}\\)	\\(\\frac{\\sqrt3}{2}\\)	\\(1\\)
\\(\\cos\\theta\\)	\\(1\\)	\\(\\frac{\\sqrt3}{2}\\)	\\(\\frac{\\sqrt2}{2}\\)	\\(\\frac{1}{2}\\)	\\(0\\)
\\(\\tan\\theta\\)	\\(0\\)	\\(\\frac{\\sqrt3}{3}\\)	\\(1\\)	\\(\\sqrt3\\)	und.

arcsin range\\[ \\sin^{-1}x\\in[-\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}] \\]

arccos range\\[ \\cos^{-1}x\\in[0,\\pi] \\]

arctan range\\[ \\tan^{-1}x\\in(-\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}) \\]

Inverse identity\\[ \\sin^{-1}x+\\cos^{-1}x=\\tfrac{\\pi}{2} \\]

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History