Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

use the method of substitution to find each of the following indefinite integrals. $$ \int v\left(\sqrt{3} v^{2}+\pi\right)^{7 / 8} d v $$

Step-by-Step Solution

Answer

\( \frac{4}{15\sqrt{3}} (\sqrt{3}v^2 + \pi)^{15/8} + C \)

Step 1: Identify the substitution

We will choose a substitution to simplify the integral. For the given integral \[ \int v(\sqrt{3} v^{2}+\pi)^{7/8} \, dv \]we notice that inside the parentheses, there is an expression that can be simplified. Let's use:\[ u = \sqrt{3}v^2 + \pi \]

Step 2: Differentiate the substitution

Differentiate the substitution to find \( du \). The derivative of \( u \) with respect to \( v \) is:\[ \frac{du}{dv} = \frac{d}{dv} (\sqrt{3}v^2 + \pi) = 2\sqrt{3}v \]Thus, \[ du = 2\sqrt{3}v \, dv \]Solving for \( v \, dv \) gives:\[ v \, dv = \frac{1}{2\sqrt{3}} \, du \]

Step 3: Substitute into the integral

Substitute \( u = \sqrt{3}v^2 + \pi \) and \( v \, dv = \frac{1}{2\sqrt{3}} \, du \) into the integral:\[ \int v(\sqrt{3} v^{2}+\pi)^{7/8} \, dv = \int (u^{7/8}) \frac{1}{2\sqrt{3}} \, du \]This simplifies to:\[ \frac{1}{2\sqrt{3}} \int u^{7/8} \, du \]

Step 4: Integrate with respect to u

Perform the integration with respect to \( u \):\[ \frac{1}{2\sqrt{3}} \int u^{7/8} \, du = \frac{1}{2\sqrt{3}} \left( \frac{u^{15/8}}{15/8} \right) + C \]Simplifying, this becomes:\[ \frac{1}{2\sqrt{3}} \times \frac{8}{15} u^{15/8} + C = \frac{4}{15\sqrt{3}} u^{15/8} + C \]

Step 5: Substitute back for v

Replace \( u \) back with \( \sqrt{3}v^2 + \pi \):\[ \frac{4}{15\sqrt{3}} (\sqrt{3}v^2 + \pi)^{15/8} + C \]

Step 6: Conclusion: Write the final answer

The indefinite integral of the given function is:\[ \int v(\sqrt{3} v^{2}+\pi)^{7/8} \, dv = \frac{4}{15\sqrt{3}} (\sqrt{3}v^2 + \pi)^{15/8} + C \]

Key Concepts

Indefinite Integrals

When we encounter an indefinite integral problem, it usually involves finding the antiderivative of a function. This means we are looking for the original function whose derivative gives the integrand. To denote an indefinite integral, we use the symbol \( \int \) followed by the integrand and the differential variable, like \( \, dv \) or \( \, dx \). An indefinite integral does not have specified limits, as opposed to a definite integral, which calculates the area under a curve between two points.

In the problem at hand, the indefinite integral is given by: \[ \int v(\sqrt{3} v^{2}+\pi)^{7/8} \, dv \] To successfully solve this, you will often need to use various techniques, one of which is the substitution method.

Integration Techniques

Integration techniques are essential in calculus to simplify and solve integrals that are not immediately obvious. Different methods include substitution, integration by parts, partial fraction decomposition, and trigonometric identities. Here, we focus on the substitution method, which is particularly useful for functions formatted with a composite nature, like polynomials nested inside trigonometric functions or other expressions.

Substitution involves changing the variable to simplify the integral. For example, we substitute part of the integrand with \( u \) to make the integral easier to handle. In this exercise, identifying the substitution is key. We notice that the expression \( \sqrt{3} v^2 + \pi \) complicates the integrand. By setting \( u = \sqrt{3} v^2 + \pi \), the expression simplifies significantly, allowing us to integrate with respect to \( u \) rather than the original variable.

Calculus Problems

In calculus, problems like the one in the exercise often involve multi-step processes that require careful handling of several concepts. These problems test your understanding of derivative and integral relationships, substitution, and reverse operations like differentiation.

To correctly solve calculus problems, it's important to:

• Understand the structure of the function you need to integrate or differentiate.
• Choose appropriate methods, such as substitution here, to break down the problem into manageable steps.
• Check your work by differentiating your antiderivative. This should yield the original integrand if done correctly.

This structured approach saves time, reduces errors, and ensures a deeper comprehension of the underlying mathematical principles.

Mathematical Substitution

Mathematical substitution is an integration technique that's particularly powerful for simplifying complex integrals. The main concept is to replace a portion of the integrand with a new variable, \( u \), turning a complicated integral into a simpler one.

In our exercise, we begin by setting \( u = \sqrt{3}v^2 + \pi \). This step is crucial as it targets the most intricate part of the integrand and simplifies it dramatically. After substitution, the variable \( v \) is no longer the focus; instead, the expression becomes something in terms of \( u \), making it easier to integrate.We then differentiate this new substitution with respect to \( v \) to express \( dv \) in terms of \( du \). This helps convert the entire original integral into a function of \( u \), leading to a straightforward integration procedure.

Finally, don't forget to substitute back to the original variable after integration is completed. This culminates in the final expression that answers the problem, making substitution a convenient and efficient tool for dealing with complex calculus problems.