Leibnitz Rule: Differentiation Under the Integral Sign
Master the fundamental theorem for differentiating integrals with variable limits - essential for JEE Advanced calculus problems.
Why Leibnitz Rule Matters for JEE Advanced
The Leibnitz Rule, also known as differentiation under the integral sign, is a powerful technique that appears in 15-20% of JEE Advanced calculus problems. Mastering this rule will help you:
- Solve complex integrals that depend on parameters
- Evaluate definite integrals with variable limits
- Tackle problems involving integral equations
- Save time on lengthy integration problems
- Gain 4-8 marks in every JEE Advanced paper
The Leibnitz Rule Formula
$$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) \cdot b'(x) - f(x,a(x)) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) dt$$
Where $a(x)$ and $b(x)$ are differentiable functions of $x$
Constant Limits
When both limits are constants: $a(x) = a$, $b(x) = b$
Simplified Formula:
$$\frac{d}{dx} \int_{a}^{b} f(x,t) dt = \int_{a}^{b} \frac{\partial}{\partial x} f(x,t) dt$$
Since $a'(x) = 0$ and $b'(x) = 0$
Example: Differentiate $I(x) = \int_{0}^{1} e^{xt} dt$
Step 1: Apply Leibnitz Rule with constant limits:
$$I'(x) = \int_{0}^{1} \frac{\partial}{\partial x} e^{xt} dt$$
Step 2: Compute partial derivative: $\frac{\partial}{\partial x} e^{xt} = te^{xt}$
Step 3: Integrate: $I'(x) = \int_{0}^{1} te^{xt} dt$
Step 4: Solve using integration by parts:
$$I'(x) = \left[\frac{te^{xt}}{x} - \frac{e^{xt}}{x^2}\right]_{0}^{1} = \frac{e^x}{x} - \frac{e^x}{x^2} + \frac{1}{x^2}$$
Variable Upper Limit
When lower limit is constant and upper limit is variable: $a(x) = a$, $b(x) = x$
Simplified Formula:
$$\frac{d}{dx} \int_{a}^{x} f(x,t) dt = f(x,x) + \int_{a}^{x} \frac{\partial}{\partial x} f(x,t) dt$$
Example: Differentiate $I(x) = \int_{0}^{x} (x-t)e^{t} dt$
Step 1: Apply Leibnitz Rule:
$$I'(x) = f(x,x) + \int_{0}^{x} \frac{\partial}{\partial x} (x-t)e^{t} dt$$
Step 2: Compute $f(x,x) = (x-x)e^x = 0$
Step 3: Partial derivative: $\frac{\partial}{\partial x} (x-t)e^{t} = e^{t}$
Step 4: Integrate: $I'(x) = \int_{0}^{x} e^{t} dt = e^x - 1$
Derivation of Leibnitz Rule
Understanding where the formula comes from helps in applying it correctly.
Step-by-step Derivation:
Step 1: Define $F(x) = \int_{a(x)}^{b(x)} f(x,t) dt$
Step 2: Use chain rule with intermediate variables:
Let $u = a(x)$, $v = b(x)$, and consider $G(x,u,v) = \int_{u}^{v} f(x,t) dt$
Step 3: Apply multivariate chain rule:
$$\frac{dF}{dx} = \frac{\partial G}{\partial x} + \frac{\partial G}{\partial u} \cdot \frac{du}{dx} + \frac{\partial G}{\partial v} \cdot \frac{dv}{dx}$$
Step 4: Compute each partial derivative:
• $\frac{\partial G}{\partial x} = \int_{u}^{v} \frac{\partial f}{\partial x} dt$
• $\frac{\partial G}{\partial u} = -f(x,u)$ (Fundamental Theorem)
• $\frac{\partial G}{\partial v} = f(x,v)$ (Fundamental Theorem)
Step 5: Substitute back to get the final formula
🚀 Problem-Solving Strategy
When to Use Leibnitz Rule:
- Integrals with parameter in integrand
- Variable limits of integration
- When direct integration is difficult
- To establish integral equations
Common Mistakes to Avoid:
- Forgetting the boundary terms
- Mixing up partial and total derivatives
- Not checking continuity/differentiability
- Applying when conditions aren't satisfied
Advanced Applications in Parts 2 & 3
Includes Gamma function evaluation, parametric integrals, and JEE Advanced level problems
📝 Quick Self-Test
Try these problems to test your understanding of Leibnitz Rule:
1. Differentiate $I(x) = \int_{0}^{x^2} \sin(xt) dt$
2. Find $\frac{d}{dx} \int_{1}^{x} \frac{e^{t}}{x} dt$
3. Evaluate $\frac{d}{dx} \int_{\sin x}^{\cos x} \sqrt{1+t^2} dt$
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