Indefinite Integration — JEE Mains Mathematics

Formulas (3)

• Clearing a root · Mixed roots of x
  $$x=t^{n},\quad dx=nt^{n-1}\,dt,\quad n=\operatorname{lcm}(\text{root orders})$$
• Two linear factors: use their ratio · Ratio substitution
  $$\int\frac{dx}{(x-a)^p(x+b)^{2-p}}=\frac{1}{(a+b)(1-p)}\left(\frac{x-a}{x+b}\right)^{1-p}+C$$
• Taking out a power of x · Power rule for a bracket
  $$\int f'(x)\,[f(x)]^{k}\,dx=\frac{[f(x)]^{k+1}}{k+1}+C,\quad k\neq-1$$

Watch out for (3)

• Given values are in x
  → Clearing a root
• Differentiate the ratio in full
  → Two linear factors: use their ratio
• A root of a power
  → Taking out a power of x

Formulas (3)

• Partial fractions · Cover-up rule
  $$\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},\quad A=\frac{pa+q}{a-b}$$
• Completing the square and x ± 1/x · Complete the square
  $$\int\frac{dx}{(x+p)^2+a^2}=\frac1a\tan^{-1}\frac{x+p}{a}+C$$
• Numerator through the denominator · Split the numerator
  $$\int\frac{A\,g(x)+B\,g'(x)}{g(x)}\,dx=Ax+B\ln|g(x)|+C$$

Watch out for (3)

• Divide before splitting
  → Partial fractions
• Match the sign to the top
  → Completing the square and x ± 1/x
• Keep the leftover constant
  → Numerator through the denominator

Formulas (2)

• Everything in tan x or cot x · Tan substitution
  $$\int\frac{dx}{a^2\sin^2x+b^2\cos^2x}=\frac1{ab}\tan^{-1}\left(\frac{a\tan x}{b}\right)+C$$
• Substituting sin x, cos x or sin x ± cos x · Sum substitution
  $$(\sin x\pm\cos x)^2=1\pm\sin2x$$

Watch out for (2)

• Cot brings a minus sign
  → Everything in tan x or cot x
• Pick t by the top
  → Substituting sin x, cos x or sin x ± cos x

Formulas (3)

• Integration by parts · Integration by parts
  $$\int u\,dv=uv-\int v\,du$$
• The pattern eˣ(f + f′) · The eˣ pattern
  $$\int e^{x}\big(f(x)+f'(x)\big)\,dx=e^{x}f(x)+C$$
• Spotting a product or quotient derivative · Quotient in reverse
  $$\int\frac{u'v-uv'}{v^2}\,dx=\frac{u}{v}+C$$

Watch out for (3)

• Signs alternate
  → Integration by parts
• Which piece is f
  → The pattern eˣ(f + f′)
• Differentiate the guess
  → Spotting a product or quotient derivative