Problem 42 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{40 d x}{x^{2}+25}$$

Step-by-Step Solution

Answer

The solution to \(\int \frac{40 dx}{x^{2}+25}\) is \(\frac{8x}{5} + C\).

Step 1: Substitution

Let's make a substitution. Let \(x = 5\tan(\theta)\). Now the next step is to differentiate x with respect to \(\theta\). This gets \(\frac{dx}{d\theta} = 5\sec^2(\theta)\), or \(dx = 5\sec^2(\theta)d\theta\).

Step 2: Transform the Integral

Substituting these into the integral, we get \(\int \frac{40 dx}{x^2 + 25} = \int \frac{40*5\sec^2(\theta) d\theta}{(5\tan(\theta))^2 + 25}\). Now simplify this to \(\int 8\sec^2(\theta) d\theta\).

Step 3: Integration

Now we need to integrate this. Recognizing \(\sec^2(\theta)\) as the derivative of \(\tan(\theta)\), the integral of 8sec²(\(\theta\)) \(d\theta\) is 8tan(\(\theta\)) + C.

Step 4: Substitute back

Substituting the original variable back, we substitute \(x = 5\tan(\theta)\) to get \(\frac{8x}{5} + C\).

Key Concepts

Substitution Method

The substitution method is a technique used in integral calculus to simplify a given integral by changing variables. It is particularly useful when the integral is difficult to solve in its original form. The goal is to transform it into an easier integral that we can solve. This transformation often involves choosing a convenient expression as the new variable, often denoted by \( u \) or \( \theta \). For example, the substitution \( x = 5\tan(\theta) \) in the given integral turns the variable \( x \) into a trigonometric function of \( \theta \).

This substitution helps because it leverages trigonometric identities that simplify the integral's components.

• Step 1: Identify a substitution that simplifies the integral.
• Step 2: Replace all occurrences of the original variable using this substitution, including differentiating to find \( dx \).

By changing variables, we turn a complex algebraic integral into a more manageable one, making integration straightforward.

Trigonometric Substitution

Trigonometric substitution is a specific type of substitution helpful for integrals involving square roots and quadratic expressions. This method uses trigonometric identities to simplify integrals by substituting trigonometric functions for algebraic expressions. For example, using trigonometric substitution, such as \( x = 5\tan(\theta) \), can simplify the expression \( x^2 + 25 \) by utilizing the Pythagorean identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

Here's a breakdown of how trigonometric substitution works:

• Choose a substitution that fits the form of the integral (e.g., for \( \sqrt{a^2 + x^2} \), consider \( x = a\tan(\theta) \)).
• Transform the integral using this substitution.

Once the substitution is made, integrate using the simplified trigonometric form, often resolving into straightforward trigonometric integrals.

Definite Integral

A definite integral represents the area under the curve of a function plotted on a graph between two specified points, often called limits. The notation \( \int_{a}^{b} f(x) \, dx \) denotes the definite integral of the function \( f(x) \) from \( a \) to \( b \).

This concept is crucial in real-life applications, such as finding the displacement in physics when velocity is a function of time, or calculating the amount of substance in chemical reactions. When solving definite integrals, the aim is to:

• First, find an antiderivative \( F(x) \) of \( f(x) \).
• Then, evaluate \( F(x) \) at the upper limit and subtract its value at the lower limit, i.e., \( F(b) - F(a) \).

Though our exercise focuses on indefinite integrals, understanding definite integrals prepares students for applications that require precise area calculations.

Integration Techniques

In integral calculus, there are several specialized methods for finding antiderivatives, known as integration techniques. These methods extend beyond basic integration to tackle complex integrals. Some common integration techniques include:

• Substitution Method: Useful when the integral can be simplified by changing variables.
• Trigonometric Substitution: Ideal for integrals involving expressions like \( x^2 + a^2 \) or other similar quadratic forms.
• Integration by Parts: Employed when the integrand can be seen as a product of two functions.

Mastering a variety of these techniques allows flexibility in approaching different integrals. Each method has its scenarios where it performs best, especially for expressions that are not easily integrable through simple antiderivation. This exercise highlights the importance of selecting the appropriate technique based on the form of the integral.