Definite Integrals & Leibniz Rule

Riemann Integral Foundation

The definite integral $\int_a^b f(x) dx$ is the limit of Riemann sums as partition mesh $\Delta x \to 0$. It measures the signed area trapped between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$.

---

🚦 Step 0 Decision Tree: Which Integral Technique to Use?

🚦 Step 0 Decision TreeStep 0: Deciding How to Evaluate a Definite Integral

Click on your mathematical problem pattern below to instantly reveal the optimal, lowest-algebra solution path:

📐Optimal Solution Blueprint for: Definite Integral with Trigonometric Fraction

1

Step 1: Check King’s Symmetry:Apply King’s property: $I = \int_a^b f(a+b-x) dx$. Write $2I = \int_a^b [f(x) + f(a+b-x)] dx$.

2

Step 2: Add Integrals:Summing the integrands usually cancels the denominator completely, leaving $\int_a^b 1 dx = b - a$.

3

Step 3: Solve for I:$2I = b - a \implies I = \frac{b - a}{2}$. Done in 3 lines without finding an antiderivative!

⚡ Pro-Tip / Exam Shortcut: For $\int_0^\pi x f(\sin x) dx$, King’s rule instantly eliminates the linear $x$ term, converting it to $\frac{\pi}{2} \int_0^\pi f(\sin x) dx$.

---

1. Fundamental Properties of Definite Integrals

#	Property Name	Mathematical Formulation
P1	Dummy Variable Invariance	${\int }_a^{b}f(x)dx={\int }_a^{b}f(t)dt$
P2	Limit Reversal	${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
P3	Interval Additivity	${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$
P4	King's Property	${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$
P5	Even/Odd Function Rule	${\int }_{-a}^{a}f(x)dx=\{\begin{array}{ll}2{\int }_0^{a}f(x)dx& \text{if }f(-x)=f(x)\\ 0& \text{if }f(-x)=-f(x)\end{array}$
P6	Periodic Function Integral	If $f(x+T)=f(x)$: ${\int }_0^{nT}f(x)dx=n{\int }_0^{T}f(x)dx$, and ${\int }_a^{a+T}f(x)dx={\int }_0^{T}f(x)dx$

---

2. Master Problem Archetype Recipe: King's Symmetry

Step 1

Identify Domain, Symmetry & Boundary Conditions

Inspect the equation or integral for hidden domain traps ($\sqrt{u} \ge 0, \ln(u) > 0, \text{denominator} \neq 0$) and parity ($f(-x) = \pm f(x)$):

$$\text{Dom}(f) = \{x \in \mathbb{R} : u(x) > 0\} \qquad\text{and check parity: }\quad f(-x) = \pm f(x)$$

↓

Step 2

Apply Canonical Invariant / Substitution Transformation

Deploy King's symmetry $x \mapsto a+b-x$, Euler polar representation $z = r e^{i\theta}$, or trigonometric/algebraic substitution:

$$I = \int_a^b f(a+b-x) dx \implies 2I = \int_a^b [f(x) + f(a+b-x)] dx \qquad\text{or}\qquad t = x \mp \frac{1}{x}$$

↓

Step 3

Closed-Form Extraction & Sanity Verification

Integrate or solve the reduced system and check extreme boundary values (e.g. $n=1, x=0$) to eliminate false branches:

$$I = \frac{b - a}{2} \qquad\text{Verify with special test case:}\quad \lim_{x\to 0^+} f(x) = L$$

---

3. Leibniz Rule for Differentiation Under the Integral Sign

Generalized Leibniz Integral Rule

If $u(x)$ and $v(x)$ are differentiable functions and $f(x, t)$ along with $\frac{\partial f}{\partial x}$ are continuous: $$\frac{d}{dx} \left[ \int_{u(x)}^{v(x)} f(x, t) dt \right] = f(x, v(x)) \cdot v'(x) - f(x, u(x)) \cdot u'(x) + \int_{u(x)}^{v(x)} \frac{\partial f}{\partial x}(x, t) dt$$

---

4. 4-Tier Scaffolded Hint Practice Drill

💡 4-Tier Scaffolded Hint SystemJEE Advanced Master Problem: Integral Equation with Leibniz Rule

Try solving with Hint 1 before unlocking deeper tiers!

🟡 Tier 1: Core Mathematical Principle & Symmetry

Recognize that the upper limit is variable $x$. Differentiating both sides using Leibniz Rule converts the integral equation into a differential equation.

🟠 Tier 2: The Opening Move & Canonical Substitution

Let $f(x) = x^2 + \int_0^x e^{-t} f(x - t) dt$. First substitute $u = x - t$ to pull $e^{-x}$ outside the integral.

🔵 Tier 3: Intermediate Milestone Equation

Multiply by $e^x$: $e^x f(x) = x^2 e^x + \int_0^x e^u f(u) du$. Differentiate both sides with respect to $x$.

🟢 Tier 4: Complete Rigorous Solution & Answer

After differentiation: $e^x f(x) + e^x f'(x) = (2x + x^2)e^x + e^x f(x) \implies f'(x) = x^2 + 2x$. Integrating with $f(0) = 0$ gives $f(x) = \frac{x^3}{3} + x^2$.

---

5. Formula Danger Zones & Constraint Checklist

📑 Quick Revision Matrix & Danger ZonesDefinite Integral & Leibniz Breakdown Conditions

Every governing formula with explicit mathematical conditions and failure danger zones:

Concept / Identity	Formula / Equation	Prerequisites	Where this Formula FAILS / Danger Zone
King's Property	$\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$	$x \in [a, b]$	Fails if $f$ has non-integrable singularities within $(a, b)$. Useful primarily when $f(x) + f(a+b-x)$ cancels denominators.
Leibniz Integral Derivative	$\frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = f(x,v)v' - f(x,u)u' + \int_u^v \frac{\partial f}{\partial x}dt$	$u, v \in C^1, \frac{\partial f}{\partial x} \in C^0$	Fails if limits $u(x), v(x)$ are discontinuous or if $\partial f/\partial x$ is not uniformly continuous on the domain.
Logarithmic Power Expansion	$\ln(x^2) = 2 \ln|x|$	$x \in \mathbb{R} \setminus \{0\}$	Writing $2\ln(x)$ without the modulus fails for all negative $x < 0$, stripping half the real domain!
Radical Product Identity	$\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$	$a \ge 0 \text{ or } b \ge 0$	Fails in $\mathbb{C}$ when both $a < 0$ and $b < 0$: $\sqrt{-1}\sqrt{-1} = i^2 = -1 \neq \sqrt{(-1)(-1)} = +1$.
AM-GM Inequality	$\frac{\sum_{i=1}^n a_i}{n} \ge \sqrt[n]{\prod_{i=1}^n a_i}$	$a_i > 0 \; (\forall i)$	Fails completely if any variable $a_i$ is negative or if equality is assumed without checking $a_1 = a_2 = \dots = a_n$.
Inverse Tangent Addition	$\arctan x + \arctan y = \arctan\left(\frac{x+y}{1-xy}\right)$	$x y < 1$	If $x > 0, y > 0$ and $x y > 1$, formula requires $+\pi$. If $x < 0, y < 0$ and $x y > 1$, formula requires $-\pi$.