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Integral Properties & Common Intergrals

Linearity of Integration​

  • Additivity: ∫ab[f(x)+g(x)] dx=∫abf(x) dx+∫abg(x) dx\int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx
  • Scalar Multiplication: ∫abcf(x) dx=c∫abf(x) dxwhere c is a constant\int_{a}^{b} cf(x) \, dx = c \int_{a}^{b} f(x) \, dx \quad \text{where } c \text{ is a constant}

Integrals over Adjacent Intervals​

  • Combining Intervals: ∫abf(x) dx=∫acf(x) dx+∫cbf(x) dxfor any value c\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx \quad \text{for any value } c
  • Zero Integral: ∫aaf(x) dx=0\int_{a}^{a} f(x) \, dx = 0

Constants in Integrals​

  • Constant Function: ∫abc dx=c(b−a)where c is a constant\int_{a}^{b} c \, dx = c(b - a) \quad \text{where } c \text{ is a constant}

Inequality Properties​

  • Order Preservation: If f(x)≥g(x)f(x) \geq g(x) on [a,b][a, b]: ∫abf(x) dx≥∫abg(x) dx\int_{a}^{b} f(x) \, dx \geq \int_{a}^{b} g(x) \, dx
  • Non-Negativity: If f(x)≥0f(x) \geq 0 on [a,b][a, b]: ∫abf(x) dx≥0\int_{a}^{b} f(x) \, dx \geq 0
  • Bounding the Integral: If m≤f(x)≤Mm \leq f(x) \leq M on [a,b][a, b]: m(b−a)≤∫abf(x) dx≤M(b−a)m(b - a) \leq \int_{a}^{b} f(x) \, dx \leq M(b - a)

Absolute Value Inequality​

  • Integral Absolute Value: ∣∫abf(x) dx∣≤∫ab∣f(x)∣ dx\left| \int_{a}^{b} f(x) \, dx \right| \leq \int_{a}^{b} |f(x)| \, dx

Common Integrals​

Basic Power and Exponential Functions​

  • Constant Function: ∫k dx=kx+c\int k \, dx = kx + c
  • Power Function (n ≠ -1): ∫xn dx=1n+1xn+1+c\int x^n \, dx = \frac{1}{n+1}x^{n+1} + c
  • Exponential Function: ∫eax dx=1aeax+c\int e^{ax} \, dx = \frac{1}{a}e^{ax} + c

Logarithmic and Reciprocal Functions​

  • Reciprocal Function: ∫1x dx=ln⁡∣x∣+c\int \frac{1}{x} \, dx = \ln|x| + c
  • Logarithmic Function: ∫ln⁡(ax) dx=xln⁡(ax)−x+c\int \ln(ax) \, dx = x \ln(ax) - x + c

Trigonometric Functions​

  • Cosine: ∫cos⁡(u) du=sin⁡(u)+c\int \cos(u) \, du = \sin(u) + c
  • Sine: ∫sin⁡(u) du=−cos⁡(u)+c\int \sin(u) \, du = -\cos(u) + c
  • Secant Squared: ∫sec⁡2(u) du=tan⁡(u)+c\int \sec^2(u) \, du = \tan(u) + c

Inverse Trigonometric Functions​

  • Arcsine (Inverse Sine): ∫1a2−u2 du=sin⁡−1(ua)+c\int \frac{1}{\sqrt{a^2 - u^2}} \, du = \sin^{-1}\left(\frac{u}{a}\right) + c
  • Arctangent (Inverse Tangent): ∫1a2+u2 du=1atan⁡−1(ua)+c\int \frac{1}{a^2 + u^2} \, du = \frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right) + c

Trigonometric Identities Involving Reciprocals​

  • Secant: ∫sec⁡(u) du=ln⁡∣sec⁡(u)+tan⁡(u)∣+c\int \sec(u) \, du = \ln|\sec(u) + \tan(u)| + c
  • Cosecant: ∫csc⁡(u) du=−ln⁡∣csc⁡(u)+cot⁡(u)∣+c\int \csc(u) \, du = -\ln|\csc(u) + \cot(u)| + c
  • Cosecant Squared: ∫csc⁡2(u) du=−cot⁡(u)+c\int \csc^2(u) \, du = -\cot(u) + c

Hyperbolic Functions​

  • Hyperbolic Sine (sinh): ∫sinh⁡(u) du=cosh⁡(u)+c\int \sinh(u) \, du = \cosh(u) + c
  • Hyperbolic Cosine (cosh): ∫cosh⁡(u) du=sinh⁡(u)+c\int \cosh(u) \, du = \sinh(u) + c

Integration by Substitution​

For a function f(u)f(u) where u=g(x)u = g(x), the substitution method can be used. For example:

  • For f(u)=sec⁡(u)tan⁡(u)f(u) = \sec(u) \tan(u), and u=g(x)u = g(x): ∫f(u) du=∫sec⁡(g(x))tan⁡(g(x))g′(x) dx\int f(u) \, du = \int \sec(g(x)) \tan(g(x)) g'(x) \, dx

Integral of a Rational Function​

  • Simple Rational Function: ∫1ax+b dx=1aln⁡∣ax+b∣+c\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln|ax + b| + c

Integrals Involving Inverse Hyperbolic Functions​

  • Inverse Hyperbolic Sine: ∫1u2+a2 du=ln⁡(u+u2+a2)+c\int \frac{1}{\sqrt{u^2 + a^2}} \, du = \ln(u + \sqrt{u^2 + a^2}) + c