What is the Leibniz integral rule?

Leibniz integral rule

Leibniz integral rule is well known as the differentiation under the integral sign rule. It provides a formula to solve a definite integral that has functions of the differential variable as its limits.

Differentiation of an integral can be calculated without integrating the equation using this rule.

Formula

Consider the following function:

According to the Leibniz integral rule, differentiating the function under the integral sign gives the following formula:

Note: Ensure that the function is of two variables, and is integrated with respect to with respect toone variable while differentiated with respect to the other variable.

Example 1

Consider the following function:

Calculate the required parameters to apply the rule:

Differentiating the function and replacing values in the Leibniz integral rule gives us the following:

It solves and concludes with the integral below:

Example 2

An example with infinity in the upper bound can also be solved using the Leibniz rule. The formula gets shortened for the following questions:

Consider the following question:

Differentiate the given equation using the Leibniz rule:

The expression evaluates the following:

Given the following equation: $f(x) = \int _{\frac{\pi^2}{16}}^{x^2}\left(\frac{sinx \cdot sin \sqrt{\theta} } {1 + cos^2 \sqrt{\theta}} \right) d\theta$

Find the value of $f'( \frac{\pi}{2})$ .

$\pi$

$-\pi$

$2\pi$

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