Indefinite Integration — NDA Maths

Formulas (10)

• Antiderivative and the Constant of Integration · Indefinite integral
  $$\int f(x)\,dx = F(x) + C \quad\text{where}\quad F'(x) = f(x)$$
• The Standard-Formula Table · Power rule (the most-used row)
  $$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
• Linearity — Integrate Term by Term · Linearity of the integral
  $$\int\big(a\,f(x)+b\,g(x)\big)\,dx = a\!\int\! f(x)\,dx + b\!\int\! g(x)\,dx$$
• Simplify the Integrand First · The collapse identity
  $$e^{\ln u} = u \qquad e^{k\ln x} = x^{k}$$
• Exponential Bases — a to the x · Exponential base rule
  $$\int a^x\,dx = \dfrac{a^x}{\ln a} + C$$
• Completing the Square for Quadratic Denominators · Arctan standard form
  $$\int \dfrac{dx}{x^2+k^2} = \dfrac{1}{k}\tan^{-1}\!\Big(\dfrac{x}{k}\Big) + C$$
• The e-to-the-x Times f-plus-f-prime Pattern · Reverse product rule
  $$\int e^x\big(f(x)+f'(x)\big)\,dx = e^x f(x) + C$$
• Cyclic and Paired Integrals of e-to-the-x Times Trig · The matched pair
  $$\int e^x\cos x\,dx = \tfrac{e^x(\cos x+\sin x)}{2}+C,\quad \int e^x\sin x\,dx = \tfrac{e^x(\sin x-\cos x)}{2}+C$$
• Properties of an Antiderivative · Integration and differentiation are inverse
  $$\int F'(x)\,dx = F(x) + C \qquad \dfrac{d}{dx}\!\int f(x)\,dx = f(x)$$
• Trigonometric Simplification Toolkit · The collapses you reach for most
  $$1+\cos x = 2\cos^2\tfrac{x}{2},\quad 1-\cos x = 2\sin^2\tfrac{x}{2},\quad \sqrt{1\pm\sin 2x}=|\sin x\pm\cos x|$$

Watch out for (12)

• Never drop the +C on an indefinite integral
  → Antiderivative and the Constant of Integration
• The power rule excludes $n=-1$
  → The Standard-Formula Table
• You cannot split a product or a quotient like a sum
  → Linearity — Integrate Term by Term
• Resolve the exponent/log BEFORE you integrate
  → Simplify the Integrand First
• Divide by $\ln a$, not by $a$
  → Exponential Bases — a to the x
• Factor out the leading coefficient first
  → Completing the Square for Quadratic Denominators
• The whole bracket must be $f + f'$
  → The e-to-the-x Times f-plus-f-prime Pattern
• du/dx is the integrand, not the other integral
  → Cyclic and Paired Integrals of e-to-the-x Times Trig
• Two true facts can still give a false link
  → Properties of an Antiderivative
• $1\pm\cos x$ (half-angle) vs $1\pm\cos 2x$ (power-reduction)
  → Trigonometric Simplification Toolkit
• The root of a perfect square is a MODULUS
  → Trigonometric Simplification Toolkit
• $\sec\pm\tan$ — mind which way the half-angle shifts
  → Trigonometric Simplification Toolkit

Formulas (6)

• Why Substitution Works — the Reverse Chain Rule · Substitution rule
  $$\int f\big(g(x)\big)\,g'(x)\,dx = \int f(u)\,du,\quad u=g(x)$$
• Algebraic and Composite Substitutions · Power-times-derivative shape
  $$\int \big(g(x)\big)^n\,g'(x)\,dx = \dfrac{\big(g(x)\big)^{n+1}}{n+1} + C$$
• The f-prime-over-f to Log Pattern · Log pattern
  $$\int \dfrac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$$
• Trigonometric Substitutions and Identity Reductions · The divide-by-cos-squared move
  $$\int \dfrac{dx}{a^2\sin^2 x + b^2\cos^2 x} = \int \dfrac{\sec^2 x\,dx}{a^2\tan^2 x + b^2} = \dfrac{1}{ab}\tan^{-1}\!\Big(\dfrac{a\tan x}{b}\Big) + C$$
• Rationalising a Surd Denominator · Conjugate clears the surd
  $$\dfrac{1}{\sqrt{x+a}-\sqrt{x+b}} = \dfrac{\sqrt{x+a}+\sqrt{x+b}}{a-b}$$
• Spotting a Hidden Derivative · The x-to-the-x derivative
  $$\dfrac{d}{dx}\,x^x = x^x(1+\ln x)$$

Watch out for (7)

• Every x must disappear before you integrate in u
  → Why Substitution Works — the Reverse Chain Rule
• Carry the sign from du
  → Algebraic and Composite Substitutions
• Adjust by a constant, never by a variable
  → The f-prime-over-f to Log Pattern
• Which of $a$, $b$ goes inside the arctan?
  → Trigonometric Substitutions and Identity Reductions
• A square root forces an absolute value
  → Trigonometric Substitutions and Identity Reductions
• Do not lose the 1 over (a minus b) factor
  → Rationalising a Surd Denominator
• x-to-the-x is neither a power nor an exponential
  → Spotting a Hidden Derivative

Formulas (3)

• The By-Parts Formula and LIATE · Integration by parts
  $$\int u\,dv = uv - \int v\,du$$
• Integrating a Lone Logarithm · Integral of the logarithm
  $$\int \ln x\,dx = x\ln x - x + C$$
• Products and Telescoping Cancellations · The product-rule trade
  $$\int u\,dv = uv - \int v\,du$$

Watch out for (3)

• Choosing u backwards makes it worse
  → The By-Parts Formula and LIATE
• ln x has no naive antiderivative
  → Integrating a Lone Logarithm
• Simplify disguised factors before applying parts
  → Products and Telescoping Cancellations

Formulas (4)

• Decomposition and the Cover-Up Method · Linear-factor decomposition
  $$\dfrac{p(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$$
• The Recurring 1 over x times x-to-the-n-plus-1 Family · Closed form for the family
  $$\int \dfrac{dx}{x(x^n+1)} = \dfrac{1}{n}\ln\left|\dfrac{x^n}{x^n+1}\right| + C$$
• Substitute First, Then Decompose · Trig-to-rational substitution
  $$\int \dfrac{\sin\theta\,d\theta}{f(\cos\theta)} = -\int \dfrac{du}{f(u)},\quad u=\cos\theta$$
• Express the Numerator via the Denominator and Its Derivative · Numerator as denom + derivative
  $$N(x) = A\,D(x) + B\,D'(x)\ \Rightarrow\ \int \dfrac{N}{D}\,dx = A x + B\ln|D| + C$$

Watch out for (4)

• Decompose only a PROPER fraction
  → Decomposition and the Cover-Up Method
• The 1 over n out front is easy to lose
  → The Recurring 1 over x times x-to-the-n-plus-1 Family
• Carry the minus from du, and the chain factor
  → Substitute First, Then Decompose
• Watch the sign in the denominator's derivative
  → Express the Numerator via the Denominator and Its Derivative