Indefinite Integration — MHT-CET Maths

Formulas (7)

• 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)
  $$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
• The Linear-Argument Rule (replace x with ax + b) · Linear-argument rule
  $$\int f(ax+b)\,dx = \dfrac{1}{a}\,F(ax+b) + C \quad\text{where}\quad \int f(x)\,dx = F(x)+C$$
• Linearity and Algebraic Pre-processing · Linearity of integration
  $$\int [a\,f(x) + b\,g(x)]\,dx = a\!\int f(x)\,dx + b\!\int g(x)\,dx$$
• Finding C from a Boundary Condition · Solving for the constant
  $$f(x) = F(x) + C,\qquad C = f(a) - F(a)$$
• Reconstruct the Function, Then Integrate · Composition of a function with itself
  $$(f\circ f)(x) = f\big(f(x)\big)$$
• 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 (9)

• Never drop the +C on an indefinite integral
  → Antiderivative and the Constant of Integration
• The power rule excludes $n = -1$
  → The Standard-Formula Table
• Only LINEAR insides get the 1/a shortcut
  → The Linear-Argument Rule (replace x with ax + b)
• Divide before you integrate an improper fraction
  → Linearity and Algebraic Pre-processing
• Solve for C only AFTER integrating
  → Finding C from a Boundary Condition
• Build f explicitly before integrating
  → Reconstruct the Function, Then Integrate
• $1\pm\cos x$ (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)

• The Substitution Rule · Substitution rule
  $$\int f(g(x))\,g'(x)\,dx = \int f(u)\,du,\quad u = g(x)$$
• The f'(x)/f(x) → log Pattern · Logarithmic integral
  $$\int \dfrac{f'(x)}{f(x)}\,dx = \log|f(x)| + C$$
• Power of a Function times its Derivative · Power-of-a-function rule
  $$\int [f(x)]^n\,f'(x)\,dx = \dfrac{[f(x)]^{n+1}}{n+1} + C \quad (n \neq -1)$$
• Root and Linear-Radical Substitutions · Linear-radical substitution
  $$t = \sqrt{ax+b}\ \Rightarrow\ x = \dfrac{t^2 - b}{a},\quad dx = \dfrac{2t}{a}\,dt$$
• Reciprocal and Take-out-the-Power Substitutions · The reciprocal substitution
  $$t = x \pm \dfrac{k}{x} \;\Rightarrow\; dt = \left(1 \mp \dfrac{k}{x^2}\right)dx$$
• Exponential and Special Substitutions · Exponential substitution
  $$t = e^{x}\ \Rightarrow\ dt = e^{x}\,dx$$

Watch out for (6)

• Adjust for the missing constant
  → The Substitution Rule
• Engineer the numerator into f'(x) + leftover
  → The f'(x)/f(x) → log Pattern
• n = −1 is NOT this rule
  → Power of a Function times its Derivative
• Re-express EVERY x, including dx
  → Root and Linear-Radical Substitutions
• Pick the sign of t = x ± k/x from the numerator
  → Reciprocal and Take-out-the-Power Substitutions
• Towers: substitute the INNER exponential
  → Exponential and Special Substitutions

Formulas (4)

• The Standard tan, cot, sec, cosec Integrals · The two that need the conjugate trick
  $$\int \sec x\,dx = \log|\sec x + \tan x| + C,\qquad \int \csc x\,dx = \log|\csc x - \cot x| + C$$
• Power Reduction with Pythagorean Identities · Key reduction identity
  $$\tan^2 x = \sec^2 x - 1$$
• Identity Simplification before Integrating · A workhorse collapse
  $$\tan x + \cot x = \dfrac{\sin^2 x + \cos^2 x}{\sin x\cos x} = \dfrac{2}{\sin 2x} = 2\csc 2x$$
• Simplify the Inverse-Trig Argument First · Cancel, then integrate
  $$\tan^{-1}(\tan\theta) = \theta,\quad \sin^{-1}(\sin\theta)=\theta \quad (\theta\text{ in principal range})$$

Watch out for (4)

• ∫sec and ∫cosec are NOT plain logs of sec/cosec
  → The Standard tan, cot, sec, cosec Integrals
• Keep one sec²x to pair with the tan-power
  → Power Reduction with Pythagorean Identities
• Try an identity before a substitution
  → Identity Simplification before Integrating
• Reduce the argument BEFORE integrating
  → Simplify the Inverse-Trig Argument First

Formulas (4)

• Standard Quadratic Denominator Forms · The arctan form
  $$\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + C$$
• Completing the Square · Completing the square
  $$ax^2 + bx + c = a\left(x + \dfrac{b}{2a}\right)^2 + \left(c - \dfrac{b^2}{4a}\right)$$
• Linear Numerator over a Quadratic (Numerator Split) · Numerator as derivative-of-denominator plus constant
  $$px + q = A\dfrac{d}{dx}(ax^2+bx+c) + B$$
• Partial-Fraction Decomposition · Distinct-linear decomposition
  $$\dfrac{px + q}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$$

Watch out for (5)

• x⁴ + bx² + c → try t = x ± k/x
  → Standard Quadratic Denominator Forms
• Factor the sign on x² before completing the square
  → Completing the Square
• Split the numerator BEFORE completing the square
  → Linear Numerator over a Quadratic (Numerator Split)
• Improper fraction? Divide before decomposing
  → Partial-Fraction Decomposition
• Repeated factor needs every power
  → Partial-Fraction Decomposition

Formulas (6)

• The Half-Angle (Weierstrass) Substitution · Weierstrass substitution
  $$t = \tan\dfrac{x}{2}:\quad \sin x = \dfrac{2t}{1+t^2},\ \cos x = \dfrac{1-t^2}{1+t^2},\ dx = \dfrac{2\,dt}{1+t^2}$$
• Divide by cos²x for a + b·sin²x Forms · After dividing by cos²x
  $$\int \dfrac{\sec^2 x\,dx}{A + B\tan^2 x} \xrightarrow{\,t=\tan x\,} \int \dfrac{dt}{A + Bt^2}$$
• Product of Two Shifted Sines (or Cosines) · Shifted-angle split
  $$\dfrac{1}{\sin(x-a)\sin(x-b)} = \dfrac{1}{\sin(b-a)}\big[\cot(x-b) - \cot(x-a)\big]$$
• Trig to Partial Fractions (substitute, then decompose) · The bridge
  $$\int R(\sin x)\,\cos x\,dx \;\xrightarrow{\,t=\sin x\,}\; \int R(t)\,dt \;\;(\text{now rational — decompose})$$
• The Fractional-Power tan Trick · The reduction (m + n even)
  $$\int \sin^m x\,\cos^n x\,dx \xrightarrow{\,t=\tan x\,} \int t^{m}\,(1+t^2)^{\frac{m+n}{2}-1}\,dt$$
• Numerator as Denominator + its Derivative · Decomposition of the numerator
  $$\text{num} = A\cdot\text{den} + B\cdot(\text{den})' \;\Rightarrow\; \int\dfrac{\text{num}}{\text{den}}\,dx = Ax + B\log|\text{den}| + C$$

Watch out for (6)

• Weierstrass is for a + b·sin/cos, not a + b·sin²
  → The Half-Angle (Weierstrass) Substitution
• Three sin/cos denominators, three substitutions
  → Divide by cos²x for a + b·sin²x Forms
• Split it — don't reach for Weierstrass
  → Product of Two Shifted Sines (or Cosines)
• No spare cos/sin → no bridge
  → Trig to Partial Fractions (substitute, then decompose)
• Check m + n is an even integer first
  → The Fractional-Power tan Trick
• Convert tan-fractions to sin/cos first
  → Numerator as Denominator + its Derivative

Formulas (5)

• Integration by Parts and the LIATE Rule · Integration by parts
  $$\int u\,dv = uv - \int v\,du$$
• Cyclic Integrals (Return-to-Self) · The cyclic result
  $$\int e^{ax}\sin bx\,dx = \dfrac{e^{ax}(a\sin bx - b\cos bx)}{a^2 + b^2} + C$$
• The eˣ[f(x) + f'(x)] Family · The eˣ[f + f'] shortcut
  $$\int e^x\big[f(x) + f'(x)\big]\,dx = e^x f(x) + C$$
• Integrals of √(quadratic) — Standard Results (syllabus reference) · Square root of a quadratic — the three results
  $$\int \sqrt{a^2 - x^2}\,dx = \dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a} + C$$
• Generalised (Tabular) By-Parts — a shortcut · Tabular by-parts series
  $$\int u\,v\,dx = u\,v_1 - u'\,v_2 + u''\,v_3 - u'''\,v_4 + \cdots$$

Watch out for (5)

• A lone log or inverse-trig still uses parts
  → Integration by Parts and the LIATE Rule
• Stop after two rounds — don't loop forever
  → Cyclic Integrals (Return-to-Self)
• Use identities to expose f + f'
  → The eˣ[f(x) + f'(x)] Family
• Syllabus result, not a current bank pattern
  → Integrals of √(quadratic) — Standard Results (syllabus reference)
• Only for a polynomial first function
  → Generalised (Tabular) By-Parts — a shortcut