Integrals revision notes

Key Concepts & Definitions

Integration (Anti-differentiation):

The inverse process of differentiation. If the derivative of a function F(x)F(x)F(x) is f(x)f(x)f(x) (i.e., ddxF(x)=f(x)\frac{d}{dx} F(x) = f(x)dxd​F(x)=f(x)), then F(x)F(x)F(x) is called an anti-derivative or primitive of f(x)f(x)f(x).

Indefinite Integral:

The formula that gives all anti-derivatives of a function is called the indefinite integral. It is denoted by ∫f(x)dx=F(x)+C\int f(x) dx = F(x) + C∫f(x)dx=F(x)+C, where CCC is the arbitrary constant of integration representing a family of parallel curves.

Integrand & Variable of Integration:

In the expression ∫f(x)dx\int f(x) dx∫f(x)dx, f(x)f(x)f(x) is the integrand and xxx is the variable of integration.

Area Function:

Defined as A(x)=∫axf(t)dtA(x) = \int_a^x f(t) dtA(x)=∫ax​f(t)dt. It represents the area of the region bounded by the curve y=f(t)y = f(t)y=f(t), the ttt-axis, and the ordinates at t=at=at=a and t=xt=xt=x.

First Fundamental Theorem of Integral Calculus:

Let fff be a continuous function on the closed interval [a,b][a, b][a,b] and A(x)A(x)A(x) be the area function. Then A′(x)=f(x)A'(x) = f(x)A′(x)=f(x), for all x∈[a,b]x \in [a, b]x∈[a,b].

Second Fundamental Theorem of Integral Calculus:

Let fff be a continuous function on the closed interval [a,b][a, b][a,b] and FFF be an anti-derivative of fff. Then the definite integral is evaluated as ∫abf(x)dx=[F(x)]ab=F(b)−F(a)\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)∫ab​f(x)dx=[F(x)]ab​=F(b)−F(a).

Methods of Integration

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.

Properties of Definite Integrals

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 ($f(-x) = -f(x)$).

JEE Advanced Specific Concepts

• 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 $\tan(x/2) = t \implies dx = \frac{2dt}{1+t^2}$, $\sin x = \frac{2t}{1+t^2}$, $\cos x = \frac{1-t^2}{1+t^2}$.

Conditions & Limitations

• 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.

Previous Year JEE Topics

• 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 $\int e^x [f(x) + f'(x)] dx$.

Standard Derivations & Step-by-Step Problem Solving

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 $ax^2+bx+c$.

Top 10 JEE MCQ Traps

1. Misconception → Correct Understanding Misconception: Substituting $x^2 = t$ in definite integrals without adjusting bounds if $x$ goes through negative values (e.g., limits from $-1$ to $1$). Correct Understanding: When setting $x^2 = t$, you must split the integral at $x=0$ because $x = \sqrt{t}$ for $x>0$ but $x = -\sqrt{t}$ for $x<0$.
2. Misconception → Correct Understanding Misconception: Believing $\int \frac{1}{x} dx$ is simply $\log(x)$. Correct Understanding: The correct anti-derivative is $\log|x| + C$. Missing the modulus traps you when the domain involves negative numbers.
3. Misconception → Correct Understanding Misconception: Applying partial fractions directly to $\frac{x^3}{x^2 - 1}$. Correct Understanding: The function is improper. You MUST perform polynomial long division first to write it as $x + \frac{x}{x^2 - 1}$ before partial fractions.
4. Misconception → Correct Understanding Misconception: Using the formula $\int e^x [f(x) + f'(x)] dx$ blindly without verifying the sign exactly. Correct Understanding: Often the trap is written as $e^{-x}[f(x) - f'(x)]$. The actual rule for $e^{-x}$ is $\int e^{-x} [f(x) - f'(x)] dx = -e^{-x}f(x) + C$.
5. Misconception → Correct Understanding Misconception: Forgetting that $\int_{-a}^a f(x) dx = 0$ ONLY applies if $f(-x) = -f(x)$ perfectly over the entire domain $[-a, a]$. Correct Understanding: If $f(x)$ is piecewise, or undefined at $x=0$ (like $\frac{1}{x}$), odd symmetry might not save you from a non-integrable divergence.
6. Misconception → Correct Understanding Misconception: Taking $\sqrt{\cos^2 x} = \cos x$ everywhere. Correct Understanding: $\sqrt{\cos^2 x} = |\cos x|$. In intervals like $[\pi/2, \pi]$, $\cos x$ is negative, so $|\cos x| = -\cos x$.
7. Misconception → Correct Understanding Misconception: Finding the area bounded by curves using simple definite integral $\int_a^b (f(x) - g(x)) dx$ without checking for intersection points inside $[a, b]$. Correct Understanding: If the curves cross, the "top" curve changes. You must find all roots of $f(x) = g(x)$ in $[a,b]$ and split the integral, taking the absolute value of each segment.
8. Misconception → Correct Understanding Misconception: Mixing up King's Rule limits: $\int_a^b f(x) dx = \int_a^b f(a-x) dx$. Correct Understanding: The correct King's Rule is $f(a+b-x)$. It only becomes $f(a-x)$ when the lower limit is explicitly $0$.
9. Misconception → Correct Understanding Misconception: Applying ILATE blindly and setting a trigonometric function as the second function $v$ when it doesn't have an easily known integral (like $\sin^{-1} x$). Correct Understanding: The "I" (Inverse Trig) and "L" (Log) come first in ILATE precisely because they don't have basic integration formulas. They must be $u$ (differentiated), and you often use $1 \cdot dx$ as the $v$ component.
10. Misconception → Correct Understanding Misconception: Forgetting the arbitrary constant $C$ when checking whether two answers from different integration methods are equivalent. Correct Understanding: Indefinite integrals are not unique functions; they are families of functions. Two completely different looking results (like $-\frac{\cos 2x}{2}$ and $\sin^2 x$) are entirely equivalent because they differ by a constant ($1/2$).