Problem 27 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \sqrt{\tan x} \sec ^{2} x d x$$

Step-by-Step Solution

Answer

The integral \(\int \sqrt{\tan x} \sec ^{2} x d x = \frac{2}{3}\tan^{\frac{3}{2}} x+ C\).

Step 1: Recognize the suitable substitution

In this case, one can see that the derivative of \(\tan x\) is \(\sec^2 x\), which is present in the integral. Thus, a suitable substitution would be to set \(u = \tan x\).

Step 2: Substitute the value of \(u\)

Replace \(\tan x\) with \(u\). Thus, the integration changes to \(\int \sqrt{u} du\). Also, note that \(du = \sec^2 x dx\).

Step 3: Solve the integral

The integral at this point is an integral of power one half, and can be easily solved using the power rule for integrals, \(\int x^n dx = \frac{x^{n+1}}{n+1}\). Applying this rule one gets, \[\frac{2}{3}u^{\frac{3}{2}}+ C\] where \(C\) is the constant of integration.

Step 4: Back substitute \(u\) with \(\tan x\)

At the last step, we replace \(u\) back with \(\tan x\) to get\[\frac{2}{3}\tan^{\frac{3}{2}} x+ C\]

Key Concepts

Calculus

Calculus is an advanced branch of mathematics that deals with change and motion. It's divided into two main parts: differential calculus, which concerns the rate at which quantities change, and integral calculus, which focuses on the accumulation of quantities. An essential concept in integral calculus is the indefinite integral, which represents the antiderivative of a function and is a way to reverse differentiation.

Understanding calculus is key in many scientific fields because it allows for the modeling and solving of problems involving dynamic systems. When we calculate the area under a curve or solve equations of motion, we are using integral calculus to make predictions and understand natural phenomena.

Indefinite Integral

The indefinite integral of a function is a fundamental concept in calculus that represents a general form of antiderivatives. The process of finding the indefinite integral is also known as integration. When you integrate a function, you are finding all possible antiderivatives of that function.

For instance, the integral of a function like \( f(x) \) with respect to \( x \) is denoted as \( \int f(x) dx \). This expression does not just represent a single solution, but rather a family of functions that differ by a constant, known as the constant of integration (\( C \)). Understanding this concept is crucial for solving problems that require reversing differentiation to find original functions.

U-Substitution

The u-substitution method, frequently just referred to as substitution, is a technique used in integral calculus to simplify the process of integration. It's particularly useful when you encounter a complex integral that is difficult to solve directly.

Here's how it works in a nutshell:

• Identify a part of the integral that, when substituted with a variable (commonly 'u'), will simplify the integral.
• Express the rest of the integral (including \( dx \)) in terms of this new variable \( u \).
• Integrate with respect to \( u \), which should be simpler.
• Substitute back the original variable to express the antiderivative in terms of the original variable.

The exercise given is a perfect example of this method and shows its practical utility in solving integrals that might not be immediately approachable through standard rules.

Power Rule for Integrals

The power rule for integrals is a basic, yet powerful, technique in calculus for finding the indefinite integral of a function that can be expressed as a power of \( x \). The rule states that for a function of the form \( x^n \), where \( n \) is a real number other than -1, the integral is \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \). The variable \( C \) represents the constant of integration.

In applying this rule, it's important to remember that it doesn't work when \( n = -1 \) because that case corresponds to the integral of \( \frac{1}{x} \), which is \( ln|x| \). Nonetheless, the power rule significantly simplifies many integration problems and is essential for solving them efficiently, as we saw in the given exercise using u-substitution.