Leibniz Rule | JEE Main Calculus Crash Course | Mathematicon

Leibniz Rule: Differentiation Under the Integral Sign

Hello JEE aspirants! Leibniz rule might seem intimidating at first glance, but it's a powerful tool to solve definite integration problems, especially those involving variable limits. Mastering this rule can significantly boost your problem-solving speed and accuracy in the JEE Main exam. Let's dive in!

What is Differentiation Under the Integral Sign?

Imagine you have a definite integral where the limits of integration are functions of $x$ , and the integrand (the function inside the integral) might also depend on $x$ . The Leibniz rule provides a way to find the derivative of such an integral with respect to $x$ . In simpler terms, it allows us to "differentiate under the integral sign."

The intuition behind this lies in understanding how changes in the limits of integration and the integrand itself affect the overall value of the integral. Consider a simple analogy: imagine calculating the area under a curve. If you slightly shift the boundaries of the area or slightly change the shape of the curve, the area itself changes. Leibniz rule gives us a precise way to calculate the rate of this change.

Leibniz Integral Rule Formula

Here's the general formula for the Leibniz rule:

\frac{d}{dx}\left[\int_{a(x)}^{b(x)} f(t) \, dt\right] = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)

Let's break this down:

$f(t)$ is the integrand (the function we're integrating). We assume $f(t)$ is continuous.
$a(x)$ and $b(x)$ are the lower and upper limits of integration, respectively. These are functions of $x$ .
$f(b(x))$ means we substitute the upper limit $b(x)$ into the function $f(t)$ .
$b'(x)$ is the derivative of the upper limit with respect to $x$ .
$f(a(x))$ means we substitute the lower limit $a(x)$ into the function $f(t)$ .
$a'(x)$ is the derivative of the lower limit with respect to $x$ .

Explanation: The derivative of the integral is equal to the function evaluated at the upper limit, multiplied by the derivative of the upper limit, minus the function evaluated at the lower limit, multiplied by the derivative of the lower limit. This accounts for how changing the limits affects the integral's value. For example, consider $\frac{d}{dx}\left[\int_{x}^{x^2} t^2 \, dt\right]$ . Using the formula, we have $f(t) = t^2$ , $a(x) = x$ , $b(x) = x^2$ . So, $f(b(x)) = (x^2)^2 = x^4$ , $b'(x) = 2x$ , $f(a(x)) = x^2$ , and $a'(x) = 1$ . Therefore, the derivative is $x^4 \cdot 2x - x^2 \cdot 1 = 2x^5 - x^2$ .

Derivation (Optional): While not essential for solving problems, understanding the derivation can provide deeper insight. Let $F(t)$ be the antiderivative of $f(t)$ , so $F'(t) = f(t)$ . Then, by the Fundamental Theorem of Calculus:

\int_{a(x)}^{b(x)} f(t) \, dt = F(b(x)) - F(a(x))

Now, differentiate both sides with respect to $x$ using the chain rule:

\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} f(t) \, dt \right] = \frac{d}{dx} \left[ F(b(x)) - F(a(x)) \right] = F'(b(x)) \cdot b'(x) - F'(a(x)) \cdot a'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)

Special Case: Only Upper Limit Varies

A common scenario is when only the upper limit of integration is a function of $x$ , and the lower limit is a constant (say, $a$ ). In this case, the formula simplifies to:

\frac{d}{dx}\left[\int_{a}^{x} f(t) \, dt\right] = f(x)

Explanation: Since $a$ is a constant, $a'(x) = 0$ , and the second term in the general Leibniz rule disappears. This is a direct consequence of the Fundamental Theorem of Calculus.

If instead the lower limit varies and the upper limit is a constant, then the formula is: $\frac{d}{dx}\left[\int_{x}^{b} f(t) \, dt\right] = -f(x)$ .

Tips for Solving Problems

Identify $f(t)$ , $a(x)$ , and $b(x)$ correctly: This is the most crucial step. Pay close attention to which variable you're integrating with respect to ( $t$ in this case) and which variable the limits depend on ( $x$ ).
Differentiate carefully: Ensure you differentiate $a(x)$ and $b(x)$ correctly. Simple differentiation errors can lead to wrong answers.
Simplify the expression: After applying the Leibniz rule, simplify the resulting expression as much as possible. This will often reveal a more manageable form.
Check for special cases: If one or both limits are constants, use the simplified version of the Leibniz rule.

For JEE problems, look for integrals where direct evaluation is difficult. Leibniz rule can often provide a shortcut by transforming the integral into a simpler expression.

Common Mistakes to Avoid

Forgetting to multiply by $a'(x)$ or $b'(x)$ : This is a very common mistake. Remember that the derivatives of the limits are essential parts of the formula.
Incorrectly substituting $b(x)$ or $a(x)$ into $f(t)$ : Make sure you replace the integration variable $t$ with the appropriate limit function.
Applying the rule when the integrand is not continuous: The Leibniz rule requires the integrand to be continuous. If there are discontinuities, the rule may not apply directly.
Confusing the variable of integration with the variable of differentiation: Always clearly identify which variable is being integrated and which variable you are differentiating with respect to.
Assuming $f(x)$ is the antiderivative: The function $f(t)$ is the integrand, *not* necessarily the antiderivative. Be careful not to confuse these!

JEE-Specific Tricks

Recognize patterns: Some JEE problems are designed to test your recognition of the Leibniz rule pattern. Practice identifying these problems quickly.
Combine with other techniques: Leibniz rule is often used in conjunction with other integration techniques like substitution or integration by parts. Be prepared to combine different methods.
Look for hidden derivatives: Sometimes, the derivative you need might be hidden within the problem. Rearrange the expression to make the derivative more apparent.

Leibniz rule is a valuable tool for JEE Main. Practice applying it to a variety of problems, and you'll be well-prepared to tackle these types of questions with confidence. Good luck!