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Calculus

Integration

Definite and indefinite integrals, properties, common integrals, integration by parts, and trigonometric integrals.

Integration is the inverse of differentiation and the way to sum infinitesimal contributions: area under a curve for the definite integral, the antiderivative family for the indefinite one. The Fundamental Theorem of Calculus links the two: a definite integral is evaluated by finding any antiderivative and taking the difference at the endpoints. The rules and tables below cover the standard integrals plus two techniques (integration by parts, case-based reduction for trigonometric integrands) for forms not in the table. Integration is linear and monotone. Unlike differentiation, it has no general product, quotient, or chain rule, so a working table of common forms is needed.

ab∫ f(x) dxxyf(x)
The definite integral of f(x) dx from a to b is the signed area between the curve and the x-axis.

Definition

A definite integral is a Riemann limit of rectangle sums. An indefinite integral is the family of antiderivatives, differing by a constant.

Definite Integral: Suppose f(x) is continuous on [a,b]. Divide [a,b] into n subintervals of width

  △ x  \text{ }\text{ }\triangle\text{ }x\text{ }\text{ }

and choose

  xi∗  \text{ }\text{ }x_{i}^{*}\text{ }\text{ }

from each interval.

The definite integral is the limit of Riemann sums:

 then ∫abf(x)dx=lim⁡n→∞f(xi∗)△ x.\text{ }then\text{ }\int_{a}^{b} f\left(x\right)dx=\lim_{n \rightarrow \infty}f\left(x_{i}^{*}\right)\triangle\text{ }x.

Anti-Derivative: An anti-derivative of f(x) is a function, F(x), such that F’(x)=f(x).

Indefinite Integral:

The indefinite integral, with constant c:

   ∫f(x)dx=F(x)+c  \text{ }\text{ }\text{ }\int f\left(x\right)dx=F\left(x\right)+c\text{ }\text{ }

where F(x) is anti-derivative of f(x).

Peroprtes

Integration is linear (sums, constants, order) and monotone.

Linearity of indefinite integrals:

∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm\int g\left(x\right)dx

Same for definite integrals:

∫abf(x)±g(x)dx=∫abf(x)dx±∫abg(x)dx\int_{a}^{b}f\left(x\right)\pm g\left(x\right)dx=\int_{a}^{b}f\left(x\right)dx\pm\int_{a}^{b}g\left(x\right)dx

Zero-width interval:

∫aaf(x)dx=0\int_{a}^{a}f\left(x\right)dx=0

Reversed limits flip the sign:

∫abf(x)dx=−∫baf(x)dx\int_{a}^{b}f\left(x\right)dx=-\int_{b}^{a}f\left(x\right)dx

Constants pull out:

∫cf(x)dx=c∫f(x)dx   ,c is a constant\int cf\left(x\right)dx=c\int f\left(x\right)dx \text{ }\text{ }\text{ },c\text{ }is\text{ }a\text{ }constant

Definite version:

∫abcf(x)dx=c∫abf(x)dx   ,c is a constant\int_{a}^{b} cf\left(x\right)dx=c\int_{a}^{b} f\left(x\right)dx \text{ }\text{ }\text{ },c\text{ }is\text{ }a\text{ }constant

The integration variable is a dummy:

∫abf(x)dx=∫abf(t)dt\int_{a}^{b}f\left(x\right)dx=\int_{a}^{b}f\left(t\right)dt

Absolute value bound:

∣∫abf(x)dx∣=∫ab∣f(x)∣dx\left|\int_{a}^{b}f\left(x\right)dx\right|=\int_{a}^{b}\left|f\left(x\right)\right|dx

Monotonicity. Pointwise inequality between integrands transfers to integrals:

if f(x)≥g(x) on a≤x≤b then  ∫abf(x)dx≥∫abg(x)dxif\text{ }f\left(x\right)\geq g\left(x\right)\text{ on }a\leq x\leq b\text{ }then\text{ }\text{ }\int_{a}^{b}f\left(x\right)dx\geq\int_{a}^{b}g\left(x\right)dx

Non-negative integrand:

if f(x)≥0 on a≤x≤b then ∫abf(x)dx≥0if\text{ }f\left(x\right)\geq0\text{ }on\text{ }a\leq x\leq b\text{ }then\text{ }\int_{a}^{b}f\left(x\right)dx\geq0

Bounds on the integrand transfer to bounds on the integral, scaled by b - a:

if m≤f(x)≤M on a≤x≤b m(b−a)≤∫abf(x)dx≤M(b−a)if\text{ }m\leq f\left(x\right)\leq M\text{ }on\text{ }a\leq x\leq b\text{ }m\left(b-a\right)\leq\int_{a}^{b}f\left(x\right)dx\leq M\left(b-a\right)

Common Integrals

Standard antiderivatives, with integration constant c.

Constant:

∫k dx=k x+c\int k\text{ }dx=k\text{ }x+c

Power rule, n ≠ -1:

∫xn dx=11+nxn+1+c   ,n≠−1\int x^{n}\text{ }dx=\frac{1}{1+n}x^{n+1}+c\text{ }\text{ }\text{ },n\neq-1

The case n = -1 gives the natural log:

∫x−1 dx=∫1xdx=ln⁡∣x∣+c\int x^{-1}\text{ }dx=\int\frac{1}{x}dx=\ln\left|x\right|+c

Linear-substitution form:

∫1ax+b dx=1aln⁡∣ax+b∣+c\int \frac{1}{ax+b}\text{ }dx=\frac{1}{a}\ln\left|ax+b\right|+c

Natural log, via integration by parts:

∫ln⁡ u du=u ln⁡(u)−u+c\int \ln\text{ }u\text{ }du=u\text{ }\ln\left(u\right)-u+c

Exponential:

∫eu du=eu+c\int e^{u}\text{ }du=e^{u}+c

Cosine:

∫cos⁡ u du=sin⁡ u+c\int\cos\text{ }u\text{ }du=\sin\text{ }u+c

Sine:

∫sin⁡ u du=−cos⁡ u+c\int\sin\text{ }u\text{ }du=-\cos\text{ }u+c

Secant squared:

∫sec⁡2 u du=tan⁡ u+c\int\sec^{2}\text{ }u\text{ }du=\tan\text{ }u+c

Secant-tangent product:

∫sec⁡ u tan⁡ u du=sec⁡ u+c\int\sec\text{ }u\text{ }\tan\text{ }u\text{ }du=\sec\text{ }u+c

Cosecant-cotangent product:

∫csc⁡ u cot⁡ u du=−csc⁡ u+c\int\csc\text{ }u\text{ }\cot\text{ }u\text{ }du=-\csc\text{ }u+c

Cosecant squared:

∫csc⁡2 u du=−cot⁡ u+c\int\csc^{2}\text{ }u\text{ }du=-\cot\text{ }u+c

Tangent:

∫tan⁡ u du=ln⁡∣sec⁡ u∣+c\int\tan\text{ }u\text{ }du=\ln\left|\sec\text{ }u\right|+c

Secant:

∫sec⁡ u du=ln⁡∣sec⁡ u tan⁡ u∣+c\int\sec\text{ }u\text{ }du=\ln\left|\sec\text{ }u\text{ }\tan\text{ }u\right|+c

Arctangent form:

∫1a2+u2 du=1atan⁡−1(ua)+a\int\frac{1}{a^{2}+u^{2}}\text{ }du=\frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right)+a

Arcsine form:

∫1a2+u2 du=sin⁡−1(ua)+a\int\frac{1}{\sqrt{a^{2}+u^{2}}}\text{ }du=\sin^{-1}\left(\frac{u}{a}\right)+a

Integration by Parts

Integration by parts is the integral analogue of the product rule. Use it when the integrand factors into a product whose pieces simplify under differentiation or integration. The LIATE heuristic picks u in the order Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential, so that u’ is simpler than u while v is still tractable to integrate. Applying parts twice often produces a self-referential equation solvable algebraically for the original integral.

∫u v dx=u∫v dx−∫u′(∫v dx)dx\int u\text{ }v\text{ }dx=u\int v\text{ }dx-\int u'\left(\int v\text{ }dx\right)dx

Where,

u is the function u(x)

v is the function v(x)

Trigonometic Integrals

Integrals of products of powers of sine, cosine, tangent, and secant reduce case by case: strip off one factor and convert the rest with a Pythagorean identity, then u-substitute.

For

  ∫sin⁡′′ x cos⁡′′′x dx  \text{ }\text{ }\int \sin''\text{ }x\text{ }\cos'''x\text{ }dx\text{ }\text{ }

we have the following:

1. n odd. Strip 1 since out and convert rest to cosines using

  sin⁡2 x=1−cos⁡2 x  \text{ }\text{ }\sin^{2}\text{ }x=1-\cos^{2}\text{ }x\text{ }\text{ }

, then use the substitution

  u=cos⁡ x  .\text{ }\text{ }u=\cos\text{ }x\text{ }\text{ .}

2. m odd. Strip 1 cosine out and convert rest to sines using

  cos⁡2 x=1−sin⁡2 x  \text{ }\text{ }\cos^{2}\text{ }x=1-\sin^{2}\text{ }x\text{ }\text{ }

, then use the substitution

  u=sin⁡ x  .\text{ }\text{ }u=\sin\text{ }x\text{ }\text{ .}

3. n and m both odd. Use either 1. or 2.

4. n and m both even. Use double angle and/or half angle formulas to reduce the integral formulas to reduce the integral into a foram tht can be integrated.

For

  ∫tan⁡′′ x sec⁡′′′ x dx  \text{ }\text{ }\int\tan''\text{ }x\text{ }\sec'''\text{ }x\text { }dx\text{ }\text{ }

we have the following:

1. n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using

  tan⁡2x=sec⁡2x−1  \text{ }\text{ }\tan^{2}x=\sec^{2}x-1\text{ }\text{ }

, then use the substitution

  u=sec⁡ x  .\text{ }\text{ }u=\sec\text{ }x\text{ }\text{ .}

2. m even. Strip 2 secants out and convert rest to tangents using

  sec⁡2x=1+tan⁡2x  \text{ }\text{ }\sec^{2}x=1+\tan^{2}x\text{ }\text{ }

, then use the substitution

  u=tan⁡ x  .\text{ }\text{ }u=\tan\text{ }x\text{ }\text{ .}

3. n odd and m even. Use either 1. or 2.

4. n even and m odd. Each integral will be dealt with differently.