Question

Derivatives of integrals Simplify the following expressions. $$\frac{d}{d x} \int_{1}^{x} e^{t^{2}} d t$$

Short answer

Question: Find the derivative of the integral $$\int_{1}^{x} e^{t^{2}} dt$$ with respect to x. Answer: The derivative of the given integral is $$\frac{d}{dx}\int_{1}^{x} e^{t^{2}} dt = e^{x^2}$$.

Step-by-step solution

Recall the Fundamental Theorem of Calculus Part 1

According to the Fundamental Theorem of Calculus (Part 1), If F(x) = ∫[a, x] f(t) dt, then F'(x) = f(x) In our case, the function f(t) inside the integral is e^(t^2). We want to find the derivative of the integral with respect to x.

Apply the Fundamental Theorem of Calculus

Applying the Fundamental Theorem of Calculus to our problem, we can say F'(x) = f(x) = e^(x^2)

Solution

The derivative of the given integral is: $$\frac{d}{dx}\int_{1}^{x} e^{t^{2}} dt = e^{x^2}$$

Key concepts

Derivative of an Integral

The derivative of an integral may sound tricky, but it actually highlights a powerful concept in calculus, known as the Fundamental Theorem of Calculus. This theorem connects differentiation and integration, showing how these two core concepts of calculus are intertwined.
The Fundamental Theorem of Calculus, Part 1 states that if you have an integral \( F(x) = \int_{a}^{x} f(t) \, dt \), then the derivative \( F'(x) \) is simply \( f(x) \).
In our specific example, the function inside the integral is \( e^{t^2} \) and the integration limit with respect to variable is \( x \). This means to find the derivative \( \frac{d}{dx}\int_{1}^{x} e^{t^2} \, dt \), you substitute \( x \) into \( t \) in \( f(t) \), resulting in \( e^{x^2} \).

The steps to solve these types of problems are:

• Identify the function within the integral \( f(t) \).
• Replace \( t \) with the upper limit \( x \) in the function.
• The result is the derivative \( f(x) \) of the integral.

Understanding this relationship is crucial for applying the Fundamental Theorem correctly.

Exponential Functions

Exponential functions are significant in calculus and the problem at hand involves \( e^{t^2} \), a form of an exponential function. Here, "e" represents Euler's number, approximately 2.71828, which is the base of natural logarithms.
Exponential functions have a unique property: they grow rapidly. The structure \( e^{variable} \) allows you to model scenarios like population growth or radioactive decay. In math, this growth is depicted beautifully through calculus.

When working with exponential functions in calculus, remember:

• The derivative of \( e^{u} \) is \( u'e^{u} \), where \( u \) is a function of \( t \).
• This property helps in solving integrals and changing them into derivatives effectively as seen in the given problem.
• Using exponential functions in integrals or derivatives often leads to simplified solutions that illustrate the deep relationship between growth rates and their underlying functions.

This intrinsic link between exponential functions and calculus operations like integration and differentiation makes them invaluable in mathematical modeling and analysis.

Integration Limits

Integration limits determine the range over which an integral is calculated, and they are crucial in defining the bounds of integration. In the expression given, \( \int_{1}^{x} e^{t^2} \, dt \), the limits are from 1 to \( x \), indicating we integrate starting at \( t = 1 \) up to \( t = x \).
Changing these limits affects the result of the integral, which is why understanding their role is essential. They significantly impact the application of the Fundamental Theorem of Calculus.

Key points about integration limits to remember:

• The lower limit, typically a constant like 1 in our case, is the starting point of integration.
• The upper limit, often a variable like \( x \), dictates the point up to which the function is integrated.
• When differentiating an integral whose upper limit is a variable, as per the Fundamental Theorem, we substitute this variable into the integrand.

By varying these limits, calculus provides flexibility in evaluating physical problems over different ranges, demonstrating the power of integration to model diverse phenomena.