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Chapter 5: Problem 25

Calculate. HINT: Exercise 24. $$\frac{d}{d x}\left(\int_{x}^{1} \frac{d t}{1}\right)$$

Short Answer

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Step by step solution

01

Identify the integral

Firstly, evaluating \(\int_{x}^{1} \frac{dt}{1}\). This is a simple integral, where the integrand function is \(\frac{1}{1}\), which is essentially \(1\). So, the integral is actually \(\int_{x}^{1} dt\), which is \(t\) for \([x, 1]\).
02

Use Fundamental Theorem of Calculus

By the Fundamental Theorem of Calculus, the integral \(\int_{x}^{1} dt = t\) on the interval \([x, 1]\) simplifies to \(1-x\). So, the expression now becomes \(\frac{d}{dx} (1-x)\).
03

Calculate the derivative

Finally, the derivative of \((1-x)\) with respect to \(x\) is \(-1\), which is the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives
Derivatives are a fundamental part of calculus. They measure how a function changes when its input changes. Essentially, the derivative gives the slope of the tangent line to the function at any point. For a function represented by \( f(x) \), the derivative is written as \( f'(x) \) or \( \frac{d}{dx}f(x) \).
  • The process of finding a derivative is known as differentiation.
  • When a function \( f(x) \) is increasing, its derivative \( f'(x) \) is positive.
  • If the function is decreasing, its derivative is negative.
In the given exercise, we differentiate the expression \( 1-x \) with respect to \( x \). Using basic differentiation rules, we find the derivative, which is \(-1\). This tells us how the quantity \( 1-x \) changes as \( x \) changes.
Definite Integrals
Definite integrals allow us to find the signed area under a curve between two limits. Integral calculus is used to sum up infinitely many infinitesimal quantities. The expression \( \int_{a}^{b} f(t)\, dt \) represents the definite integral of \( f(t) \) from \( t=a \) to \( t=b \).
  • The definite integral provides the net area between the curve and the x-axis, accounting for areas below the x-axis as negative and above as positive.
  • Geometrically, it measures the accumulated change of the quantity represented by the integral, within the given interval.
In the problem at hand, \( \int_{x}^{1} dt \) represents the integral from \( x \) to \( 1 \) of a constant function \( 1 \). The solution evaluates this to \( 1-x \), using the limits.
Basic Integration Techniques
Basic integration techniques are used to simplify the process of finding integrals. The integration of constant functions is one of the simplest forms of integration. When integrating a constant \( c \) over an interval \([a, b]\), the integral is \( c(b-a) \).
  • A classic rule states that \( \int c\, dt = ct + C \), where \( C \) is the constant of integration.
  • For definite integrals, we evaluate the antiderivative at the boundaries and subtract: \( F(b) - F(a) \).
The exercise applies these basic techniques to evaluate \( \int_{x}^{1} dt \) as \( t|_{x}^{1} \), which resolves to \( 1-x \). This simple result emerges from understanding and applying the basic techniques of integration to constant functions.

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