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Chapter 5: Problem 38

Evaluate. $$ \int \frac{d}{d x}(\sqrt{\tan x}) d x $$

Short Answer

Expert verified
\( \sqrt{\tan x} + C \), where \( C \) is the constant of integration.

Step by step solution

01

Identify the Integral

The problem requires us to evaluate the integral \( \int \frac{d}{dx}(\sqrt{\tan x}) \, dx \). The expression inside the integral represents the derivative of \( \sqrt{\tan x} \).
02

Apply Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if you have the integral of the derivative of a function, i.e., \( \int \frac{d}{dx}(f(x)) \, dx \), it simplifies to the function itself plus a constant of integration. Therefore, \( \int \frac{d}{dx}(\sqrt{\tan x}) \, dx = \sqrt{\tan x} + C \), where \( C \) is the constant of integration.

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Key Concepts

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

Definite Integrals
Definite integrals play a crucial role in calculus as they help calculate the area under curves between two specific points. When you evaluate a definite integral, you are primarily interested in the cumulative value of a function's behavior between these bounds. The limits of integration, often denoted by the lower and upper bounds, define precisely which segment of the curve is under consideration.

When dealing with definite integrals:
  • You need to specify the interval \[a, b\], which represents the starting and ending points of integration on the x-axis.
  • The result of a definite integral is a number, representing the total net area enclosed by the curve and the x-axis, from point a to b.
  • If the area is above the x-axis, it contributes positively, and if below, negatively.
Understanding definite integrals is essential for solving many real-world problems like calculating distances, areas, and even volumes in different contexts.
Indefinite Integrals
Indefinite integrals represent the family of all antiderivatives of a function. Unlike definite integrals, they do not have specified bounds and are characterized by the inclusion of an arbitrary constant, often denoted as C. This constant arises because differentiating different constants results in the same derivative; hence, integrating could correspond to multiple functions differing by a constant.

Here are some key points about indefinite integrals:
  • They give us a general formula or function (F(x)) whose derivative is the integrand, f(x).
  • The arbitrary constant C is pivotal in indefinite integrals because it accounts for all potential vertical shifts in the antiderivative function.
  • The notation of indefinite integrals is \( \int f(x) \, dx \), which reads as "the integral of f(x) with respect to x."
Indefinite integrals are fundamental when it comes to solving problems that involve finding original functions from known slopes or rates of change.
Derivatives
Derivatives measure the rate at which a function changes. Observing how quickly or slowly a function alters at any given point is crucial in numerous fields, like physics, engineering, and economics. When you take the derivative of a function, you're finding its instantaneous rate of change at each point.

Important highlights about derivatives include:
  • The derivative of a function \( f(x) \) is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).
  • It represents the slope of the tangent line to the curve of the function at any given point.
  • Finding derivatives helps in determining crucial aspects like velocity and acceleration in motion.
In our problem, applying the derivative to \( \sqrt{\tan x} \) and integrating it back to evaluate the integral illustrates the fundamental relationship between differentiation and integration encapsulated in the Fundamental Theorem of Calculus. This theorem elegantly bridges these two essential operations.

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