Problem 22 - Calculus for Biology and Medicine

Question

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{2 x^{2}-1}\left(e^{-2 t}+e^{2 t}\right) d t $$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = 4x(e^{-4x^2+2} + e^{4x^2-2})\).

Step 1: Identify the Integral Boundaries

The given function involves an integral with variable upper and constant lower boundary y = \(\int_0^{2x^2-1}(e^{-2t} + e^{2t}) dt\). To apply Leibniz's rule, recognize that the variable boundary is the upper limit \(u(x)=2x^2-1\).

Step 2: Differentiate Using Leibniz's Rule

According to Leibniz's rule for differentiation under the integral sign, \(\frac{d}{dx}\int_{a(x)}^{b(x)} f(t, x) \, dt = f(b(x), x)\cdot\frac{db(x)}{dx} - f(a(x), x)\cdot\frac{da(x)}{dx} + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(t, x) \, dt\). Since our lower limit is constant (0), the middle term and integral term vanish in this scenario.

Step 3: Apply Leibniz Rule - Evaluate at Upper Bound

Since the function in integral does not depend on \(x\) directly, the derivative only involves the first term of Leibniz's rule. Substitute \(t = 2x^2-1\) into the function \(e^{-2t} + e^{2t}\) to get \(e^{-2(2x^2-1)} + e^{2(2x^2-1)}\).

Step 4: Calculate the Derivative of the Upper Limit

The derivative of the upper limit \(2x^2 - 1\) with respect to \(x\) is \(\frac{d}{dx}(2x^2 - 1) = 4x\).

Step 5: Combine Results to Find \(\frac{dy}{dx}\)

Using the results from Step 3 and Step 4, compute \(\frac{dy}{dx} = (e^{-2(2x^2-1)} + e^{2(2x^2-1)})\cdot4x\). Simplified, this expression becomes \(4x(e^{-4x^2+2} + e^{4x^2-2})\).

Key Concepts

Differentiation Under the Integral Sign

Differentiation under the integral sign is a powerful technique in calculus, allowing one to find the derivative of an integral whose limits are functions of the variable of differentiation. It's especially useful when dealing with integrals where the limits, or even the integrand, depend on the variable of interest. This concept is beautifully captured in Leibniz's rule, which handles such scenarios with elegance.

Leibniz's rule provides the formula:

• \( \frac{d}{dx}\int_{a(x)}^{b(x)} f(t, x) \, dt = f(b(x), x)\cdot\frac{db(x)}{dx} - f(a(x), x)\cdot\frac{da(x)}{dx} + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(t, x) \, dt \)

When the lower boundary is constant, such as 0, the second term disappears, simplifying our calculations. In many applications, the integrand doesn't explicitly depend on \(x\), which further simplifies to calculating the first term of Liebniz's rule. This simplification helps us focus on the impact of the changing upper limit on the integral value, as seen in many calculus problems.

Understanding and applying this rule allows for the solution of complex calculus problems more intuitively and can provide insights that other methods might miss. It helps in breaking down the complexity by focusing directly on how changes in one quantity affect another.

Variable Boundary Integration

Integrating functions with variable boundaries introduces an additional layer of complexity to calculus problems. Unlike fixed boundary integrals, where limits are constants, variable boundary integrals depend on one or more variables.

In the integral \( y = \int_0^{2x^2-1}(e^{-2t} + e^{2t}) dt \), the upper boundary \(2x^2 - 1\) varies with \(x\).
This variability means that as \(x\) changes, the range of integration changes too—a typical scenario dealt with using Leibniz's rule. The interaction between \(x\) and the integral's range needs special attention, as it directly influences the integral's final value.

When approaching problems like these, it's crucial to first identify the boundaries clearly and understand how they depend on the variable in question. This step ensures correct application of formulations like Leibniz's rule, where careful substitution and differentiation occur. Grasping the dynamics of these changing boundaries better prepares you to tackle a wide range of applied and theoretical calculus problems.

Calculus Problem-Solving

In calculus, problem-solving revolves around applying known methods creatively to new situations. Mastery requires not only understanding theory but also how to apply it strategically to solve problems efficiently.

For example, finding \( \frac{dy}{dx} \) in problems like \( y = \int_0^{2x^2-1}(e^{-2t} + e^{2t}) dt \) involves:

• Recognizing the role of variable boundaries and how they necessitate different techniques than constant boundary integrals.
• Systematically applying Leibniz's rule ensures the formula aligns with the specific constraints and characteristics of the integrand and variable limits.
• Strategically substituting and differentiating terms, such as finding \( \frac{d}{dx}(2x^2 - 1) \).

Through these steps, one compiles all needed components to solve for derivatives efficiently, simplifying complex data into manageable parts. By practicing such methodologies, one becomes adept at identifying essential aspects and executing calculations with precision.