Question

use the method of substitution to find each of the following indefinite integrals. $$ \int \cos (\pi v-\sqrt{7}) d v $$

Short answer

\( \int \cos(\pi v - \sqrt{7}) \, dv = \frac{1}{\pi} \sin(\pi v - \sqrt{7}) + C \).

Step-by-step solution

Identify the Inner Function

We notice the expression inside the cosine function is \( u = \pi v - \sqrt{7} \), which we can substitute for easier integration.

Calculate the Derivative of the Inner Function

Find the derivative of \( u \) with respect to \( v \): \( \frac{du}{dv} = \pi \).

Solve for dv in terms of du

Rearrange the derivative: \( dv = \frac{du}{\pi} \).

Substitute into the Integral

Replace \( \pi v - \sqrt{7} \) with \( u \) and \( dv \) with \( \frac{du}{\pi} \), transforming the integral into \( \int \cos(u) \cdot \frac{du}{\pi} \).

Simplify the Integral

Factor out the constant \( \frac{1}{\pi} \): \( \frac{1}{\pi} \int \cos(u) \, du \).

Integrate with Respect to u

The integral of \( \cos(u) \) is \( \sin(u) + C \). So, \( \int \cos(u) \, du = \sin(u) + C \).

Substitute Back to Original Variable

Substitute \( u = \pi v - \sqrt{7} \) back into the integral, giving \( \frac{1}{\pi} \sin(\pi v - \sqrt{7}) + C \).

Key concepts

Substitution Method

The substitution method in integral calculus is a powerful technique that simplifies the integration process, especially when dealing with complex functions. It works by making a substitution, or change of variable, to transform the original integral into a simpler one. Here's how it generally works:- **Identify the Inner Function**: Look for a function inside another function, such as the expression "\( \pi v - \sqrt{7} \)" in our example. This inner function is what you will replace with a single variable, typically denoted as \( u \).- **Calculate the Derivative**: Find the derivative of the inner function with respect to the original variable. This gives us the expression for \( du \).- **Rewrite the Differential**: Solve the derivative expression to express the differential of the original variable \( dv \) in terms of \( du \). This often involves manipulating the equation to find \( dv = \frac{du}{\pi} \) in our case.- **Substitute in the Integral**: Replace all instances of the original variables and differentials with the new variable \( u \) and the corresponding differential \( du \) to simplify the integral.- **Simplify and Integrate**: Integrate with respect to the new variable \( u \), and then substitute back to express the result in terms of the original variable.Using substitution can make complex trigonometric and algebraic expressions much easier to handle in indefinite integrals.

Indefinite Integral

An indefinite integral in calculus refers to the general form of the antiderivative of a function. Unlike a definite integral, which computes the area under the curve between two points, an indefinite integral provides a family of functions meant to represent the original function's antiderivative.- **General Form**: The indefinite integral of a function \( f(x) \) is generally expressed as \( \int f(x) \, dx \, = F(x) + C \), where \( F(x) \) is an antiderivative of \( f(x) \), and \( C \) is the constant of integration.- **Constant of Integration**: \( C \) is crucial as it accounts for the fact that antiderivatives are determined up to an additive constant. This reflects the inherent ambiguity in recovering a function solely from its derivative.- **Purpose**: The process of finding an indefinite integral is designed to reverse differentiation. For instance, in our example, integrating \( \cos(u) \) with respect to \( u \) yields \( \sin(u) + C \).Grasping indefinite integrals is essential for solving a wide range of problems in physics and engineering, where instead of boundaries, the focus is on obtaining general solutions.

Trigonometric Integration

Trigonometric integration involves integrating functions that contain trigonometric terms. These can sometimes be challenging due to their complex periodic nature, but specific rules and strategies are available to simplify the process.- **Common Functions**: Functions like \( \sin(x) \), \( \cos(x) \), and their various products and powers are typical subjects of trigonometric integration. Each has specific integration rules, such as the integral of \( \cos(x) \) being \( \sin(x) + C \).- **Substitution and Simplification**: Often, trigonometric integrals are made easier through substitution, particularly when combined with angles or other functions inside the trigonometric term. For example, substituting \( u = \pi v - \sqrt{7} \) allows the usage of simpler integration rules.- **Use of Identities**: Trigonometric identities, like the Pythagorean identity and angle-sum formulas, can transform complex expressions into integrable formats. While not needed in our specific example, these identities are foundational tools.Successfully integrating trigonometric functions can aid in solving oscillatory-type problems, such as wave equations and alternating current (AC) circuit analyses, which inherently involve cosine and sine functions.