Integrals revision notes

Standard Indefinite Integrals:

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
• $\int dx = x + C$
• $\int \frac{1}{x} dx = \log|x| + C$
• $\int e^x dx = e^x + C$
• $\int a^x dx = \frac{a^x}{\log a} + C$
• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$
• $\int \csc^2 x dx = -\cot x + C$
• $\int \sec x \tan x dx = \sec x + C$
• $\int \csc x \cot x dx = -\csc x + C$
• $\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1} x + C$ (or $-\cos^{-1} x + C$)
• $\int \frac{1}{1+x^2} dx = \tan^{-1} x + C$ (or $-\cot^{-1} x + C$)

Integrals derived via Substitution:

• $\int \tan x dx = \log|\sec x| + C = -\log|\cos x| + C$
• $\int \cot x dx = \log|\sin x| + C$
• $\int \sec x dx = \log|\sec x + \tan x| + C = \log|\tan(\frac{\pi}{4} + \frac{x}{2})| + C$
• $\int \csc x dx = \log|\csc x - \cot x| + C = \log|\tan(\frac{x}{2})| + C$

Six Special Integrals (Denominator forms):

• $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log|\frac{x-a}{x+a}| + C$
• $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log|\frac{a+x}{a-x}| + C$
• $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$
• $\int \frac{dx}{\sqrt{x^2 - a^2}} = \log|x + \sqrt{x^2 - a^2}| + C$
• $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a}) + C$
• $\int \frac{dx}{\sqrt{x^2 + a^2}} = \log|x + \sqrt{x^2 + a^2}| + C$

Three Special Integrals (Numerator square root forms):

• $\int \sqrt{x^2 - a^2} dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \log|x + \sqrt{x^2 - a^2}| + C$
• $\int \sqrt{x^2 + a^2} dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \log|x + \sqrt{x^2 + a^2}| + C$
• $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$

Integration by Parts:

• $\int u v dx = u \int v dx - \int (u' \int v dx) dx$
• Classic Form: $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$

1. Method of Substitution

Transforming $\int f(x) dx$ by substituting $x = g(t)$, which yields $dx = g'(t) dt$. The integral becomes $\int f(g(t)) g'(t) dt$. JEE Tip Always remember to change the limits of integration when substituting in a definite integral to avoid having to back-substitute.

2. Integration by Partial Fractions

Used when the integrand is a rational function $\frac{P(x)}{Q(x)}$.

• Proper Rational Function: Degree of $P(x) <$ Degree of $Q(x)$.
• Improper Rational Function: Degree of $P(x) \ge$ Degree of $Q(x)$. JEE Tip Always perform polynomial long division first to make it proper: $\frac{P(x)}{Q(x)} = T(x) + \frac{P_1(x)}{Q(x)}$.

Partial Fraction Decomposition Forms:

3. Integration by Parts

Used for integrating the product of two functions. JEE Tip Use the ILATE rule to choose the 1st function ($u$): Inverse Trig, Logarithmic, Algebraic, Trigonometric, Exponential. The function occurring first in ILATE is taken as the 1st function because its derivative simplifies the expression, while the 2nd function must be easily integrable.

4. Integral of specific linear/quadratic forms

• For $\int \frac{dx}{ax^2+bx+c}$ or $\int \frac{dx}{\sqrt{ax^2+bx+c}}$: Use the method of "completing the square" for the quadratic polynomial in the denominator.
• For $\int \frac{px+q}{ax^2+bx+c} dx$ or $\int \frac{px+q}{\sqrt{ax^2+bx+c}} dx$: Express the numerator as $px+q = A \frac{d}{dx}(ax^2+bx+c) + B$. Find $A$ and $B$ by equating coefficients.

These properties are critical for simplifying definite integrals:

• $P_0$ (Dummy Variable Property): $\int_a^b f(x) dx = \int_a^b f(t) dt$
• $P_1$ (Limit Reversal): $\int_a^b f(x) dx = -\int_b^a f(x) dx$
• $P_2$ (Splitting Property): $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ JEE Tip Extremely useful for piecewise functions (e.g., modulus $|x|$, greatest integer $[x]$, fractional part $\{x\}$).
• $P_3$ (King's Rule): $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$
• $P_4$ (King's Rule - Special Case): $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ JEE Tip This is the most frequently tested property in JEE. Apply it when the denominator of the integrand remains unchanged upon replacing $x$ with $a+b-x$ or $a-x$.
• $P_5$ (Half-Limit Property): $\int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a-x) dx$
• $P_6$ (Queen's Rule): $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ if $f(2a-x) = f(x)$, and $0$ if $f(2a-x) = -f(x)$.
• $P_7$ (Even/Odd Property): $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is an EVEN function ($f(-x) = f(x)$). It evaluates to $0$ if $f(x)$ is an ODD function ($f(-x) = -f(x)$).

• Newton-Leibniz Formula (Differentiation under the Integral Sign): $\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) dt = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x)$ JEE Tip Always use this formula when you see a limit $x \to a$ with an integral in the numerator/denominator (paired with L'Hopital's rule) or when solving differential equations defined via integrals.
• Definite Integral as the Limit of a Sum: $\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n}) = \int_0^1 f(x) dx$ JEE Tip Convert summation to integration by substituting: $\frac{r}{n} \to x$, $\frac{1}{n} \to dx$, and the summation limits to $\int_{a}^{b}$ where $a = \lim_{n \to \infty} \frac{r_{min}}{n}$ and $b = \lim_{n \to \infty} \frac{r_{max}}{n}$.
• Wallis' Formula (Reduction Formula): $\int_0^{\pi/2} \sin^n x \cos^m x dx = \frac{[(n-1)(n-3)...][(m-1)(m-3)...]}{(n+m)(n+m-2)...} \times K$ Where $K = \pi/2$ if both $n$ and $m$ are even, otherwise $K = 1$.
• Periodicity in Definite Integrals: If $f(x)$ is periodic with period $T$: $\int_0^{nT} f(x) dx = n \int_0^T f(x) dx$.
• Estimation of Definite Integrals: If $m \le f(x) \le M$ for $x \in [a, b]$, then $m(b-a) \le \int_a^b f(x) dx \le M(b-a)$.
• Euler's Substitution / Half-Angle Substitution: For integrands with rational sines and cosines like $\frac{1}{a \sin x + b \cos x + c}$, substitute $\tan(x/2) = t \implies dx = \frac{2dt}{1+t^2}$, $\sin x = \frac{2t}{1+t^2}$, $\cos x = \frac{1-t^2}{1+t^2}$.

• Power Rule Continuity: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ is explicitly NOT valid for $n = -1$. For $n = -1$, the integral is $\log|x| + C$.
• FTC Requirement: The Fundamental Theorem of Calculus (I & II) strictly requires the function $f(x)$ to be continuous in the closed interval $[a, b]$. If the function is discontinuous, it must be split at the points of discontinuity using Property $P_2$.
• Domain constraints for Log/Inverse functions: The formula $\int \frac{1}{x} dx = \log|x| + C$ includes the modulus because logarithms are undefined for negative numbers. Similarly, when using trigonometric substitutions (e.g., $x = a \tan \theta$), restrict $\theta$ to the principal domains to ensure mapping is bijective.
• Partial Fractions: Can only be applied to proper rational functions. If improper, long division must be executed first.

• Missing the arbitrary constant: The indefinite integral $\int f(x)dx$ represents an infinite family of curves differing by the arbitrary constant $C$. Forgetting $C$ in indefinite integrals is a critical error. However, $C$ cancels out and is completely absent in definite integrals.
• Square root simplification: $\sqrt{x^2} = |x|$, NOT $x$. JEE Tip When evaluating definite integrals like $\int_{-1}^1 \sqrt{x^2} dx$, you must write it as $\int_{-1}^1 |x| dx$ and split it into $\int_{-1}^0 (-x)dx + \int_0^1 x dx$. Blindly writing $x$ leads to $0$ instead of the correct area ($1$).
• Integral of a Product: $\int f(x)g(x) dx \neq \int f(x)dx \times \int g(x) dx$. Always use Integration by Parts for products.
• Substitution in Definite Integrals without changing limits: A fatal error is substituting $x = g(t)$ but keeping the limits $a$ and $b$ (which belong to $x$). You must calculate the new limits $t_1 = g^{-1}(a)$ and $t_2 = g^{-1}(b)$.
• Definite integral of discontinuous functions: Using $\int_{-1}^1 \frac{1}{x^2} dx = [-\frac{1}{x}]_{-1}^1 = -1 - (1) = -2$ is completely WRONG because $1/x^2$ is discontinuous at $x=0$. Area must be positive, and this integral diverges to infinity.

• King's Rule ($P_3, P_4$) coupled with Odd/Even ($P_7$): Vast majority of JEE definite integration questions test your ability to reflect the limits using $a+b-x$ to cancel out complex numerators.
• Piecewise Function Evaluation: Area bounding involving $|x-a|$, $[x]$ (greatest integer function), and $\{x\}$ (fractional part) using the Splitting Property $P_2$.
• Newton-Leibniz formula with Limits: $\lim_{x\to 0} \frac{\int_0^x f(t) dt}{x^2}$ style questions relying heavily on L'Hopital's rule.
• Limit of a sum: Identifying $\lim_{n \to \infty} \sum$ expressions and converting them cleanly into definite integrals.
• Classic $e^x$ pattern: Questions designed to be reduced algebraically or via trig identities to the form $\int e^x [f(x) + f'(x)] dx$.

Derivation of $\int e^x (f(x) + f'(x)) dx$

1. Split the integral: $\int e^x f(x) dx + \int e^x f'(x) dx$.
2. Apply Integration by Parts to the first integral only, taking $f(x)$ as the 1st function and $e^x$ as the 2nd function.
3. $\int e^x f(x) dx = f(x)e^x - \int f'(x)e^x dx$.
4. Substitute back into step 1: $f(x)e^x - \int f'(x)e^x dx + \int e^x f'(x) dx$.
5. The integrals cancel, leaving $e^x f(x) + C$.

Step-by-Step Problem Solving for $\int \frac{px+q}{ax^2+bx+c} dx$

1. Equate numerator: $px+q = A \frac{d}{dx}(ax^2+bx+c) + B = A(2ax+b) + B$.
2. Compare coefficients of $x$: $p = 2Aa \implies A = \frac{p}{2a}$.
3. Compare constant terms: $q = Ab + B \implies B = q - \frac{pb}{2a}$.
4. Split integral into two parts: $A \int \frac{2ax+b}{ax^2+bx+c} dx + B \int \frac{dx}{ax^2+bx+c}$.
5. The first integral trivially becomes $A \log|ax^2+bx+c|$.
6. The second integral is solved by completing the square for $ax^2+bx+c$.