Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{2}^{x}\left(3 t+\cos t^{2}\right) d t$$

Short answer

The derivative \(d y / d x = 3x + \cos x^{2}\)

Step-by-step solution

Apply Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the derivative of an integral of a function is simply the original function. This could be written as \(d / d x \int_{a}^{x} f(t) d t=f(x)\), if \(f(t)\) is continuous. Here, the function to be integrated is \(f(t) = 3t + \cos t^{2}\). So, according to the theorem, if you differentiate \(y\) with respect to \(x\) you get the original function back: \(d y / d x = 3x + \cos x^{2}\).

State the final result

The derivative of \(y\) with respect to \(x\), \(d y / d x\), is equal to \(3x + \cos x^{2}\).

Key concepts

Differentiation

Differentiation is a process in mathematics that deals with finding the rate at which a function changes at any given point. It is a fundamental tool used for understanding the behavior of functions and solving problems in calculus. When you differentiate a function, you are essentially seeking the derivative, which gives you the slope of the function at any point along its curve.

In the context of our exercise, differentiation allows us to find the rate of change of the function \(y\) with respect to \(x\). Using the Fundamental Theorem of Calculus, the step-by-step solution shows that when you differentiate the integral of \(f(t) = 3t + \text{cos}(t^2)\) from 2 to \(x\), the derivative \(dy/dx\) gives you the original function evaluated at \(x\), hence \(3x + \text{cos}(x^2)\).

Integration

Integration is the inverse process of differentiation. It is used to find the total amount that accumulates when a quantity changes at a certain rate over a particular interval. While differentiation helps find the rate of change, integration helps find the 'total change' over an interval.

Integration can be thought of as the process of adding up infinite infinitesimally small pieces to find the whole. There are two main types of integrals: indefinite and definite. The indefinite integral represents the collection of all antiderivatives of a function, while the definite integral calculates the accumulation over a specific interval.

In our exercise, the definite integral \(\int_{2}^{x}(3t + \text{cos}(t^2)) dt\) represents the total accumulation of the function \(3t + \text{cos}(t^2)\) as \(t\) varies from 2 to \(x\).

Definite Integrals

Definite integrals have both start and end points, unlike indefinite integrals which represent a family of functions. They are often used to calculate areas under the curve, total quantities, and other accumulations between two bounds.

A definitive integral has the form \(\int_{a}^{b} f(x) dx\), where \(a\) and \(b\) are the lower and upper limits of integration, respectively. It calculates the net area between the function \(f(x)\) and the \(x\)-axis over the interval \([a, b]\).

In the given exercise, the definite integral from 2 to \(x\) of the function \(3t + \text{cos}(t^2)\) is the accumulation of the values of \(f(t)\) along the interval from 2 to \(x\). According to the Fundamental Theorem of Calculus, evaluating the derivative of this integral gives us a function that depicts the rate of change of this accumulated value at the upper limit, which is our variable \(x\).