Question

Find the derivative. \(\frac{d}{d t} \int_{0}^{t} \frac{\sin x}{\sqrt{1+x^{2}}} d x\)

Short answer

\(\frac{d}{dt}\int_{0}^{t} \frac{\sin x}{\sqrt{1+x^{2}}} \, dx = \frac{\sin t}{\sqrt{1+t^{2}}}\).

Step-by-step solution

Identify the Problem Type

We need to find the derivative with respect to \( t \) of an integral whose limits are from 0 to \( t \). This indicates that the Fundamental Theorem of Calculus Part 1 is applicable.

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus Part 1, if \( F(t) = \int_{a}^{t} f(x) \; dx \), then \( \frac{dF}{dt} = f(t) \). Therefore, we apply this directly to our problem.

Substitute the Function in the Fundamental Theorem Formula

Substitute \( f(x) = \frac{\sin x}{\sqrt{1+x^{2}}} \) into the formula from Step 2. Thus, the derivative \( \frac{d}{dt} \int_{0}^{t} \frac{\sin x}{\sqrt{1+x^{2}}} \; dx = \frac{\sin t}{\sqrt{1+t^{2}}} \).

Key concepts

Derivative of an Integral

The exercise involves differentiating an integral, specifically finding the derivative with respect to a variable when it's the upper limit of the integral. This is a direct application of the **Fundamental Theorem of Calculus, Part 1**. This theorem links the process of differentiation and integration. It asserts that if we have a function defined as an integral with a variable upper limit, like \( F(t) = \int_a^t f(x) \, dx \), then the derivative of \( F \) with respect to \( t \) is simply \( f(t) \).

This is powerful because it allows us to transform what might be a complex integration problem into a much simpler differentiation problem. In our exercise, the function is \( f(x) = \frac{\sin x}{\sqrt{1+x^2}} \), and according to the theorem, the derivative \( \frac{d}{dt} \) of the integral from \( 0 \) to \( t \) is simply \( f(t) = \frac{\sin t}{\sqrt{1+t^2}} \).

By understanding this concept, you can handle similar exercises efficiently, transforming integrals into derivatives without directly solving the integral itself.

Calculus

Calculus is the study of change and motion through derivatives and integrals. It's a fundamental part of mathematics that allows us to understand and describe changes whether they are gradual or sudden. Calculus is split into two major branches:

• Differential Calculus: This part studies the concept of the derivative, which represents rates of change and slopes of curves. It answers questions like, "How fast is something changing over time?"
• Integral Calculus: This part deals with integrals and accumulation of quantities, such as areas under a curve and total distance traveled.

Each branch has its distinct concepts and applications, but they are interconnected, as shown by the Fundamental Theorem of Calculus. Understanding these connections helps reveal the deeper structure of mathematics and offers powerful tools for solving real-world problems.

Integral Calculus

Integral calculus focuses on the process of integration, essentially the reverse process of differentiation. It helps us to find areas, volumes, central points, and many useful things. There are two main types of integrals:

• Definite Integrals: These compute the accumulation of a quantity over an interval, giving a numerical result. For instance, they are used to compute areas under a curve.
• Indefinite Integrals: Also known as antiderivatives, these represent a family of functions and include a constant of integration \( C \).

In our example, we worked with a definite integral, where our goal was to differentiate this integral concerning its upper limit. Through the **Fundamental Theorem of Calculus**, we see that the derivative of this integral is simply the value of the integrand at the upper limit. This reveals the powerful symbiotic relationship between the processes of differentiation and integration, making it easier to tackle complex problems by leveraging this deep mathematical insight.