Problem 24 - Calculus

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

The integral is \( \frac{4}{15\sqrt{3}}\left(\sqrt{3} v^{2} + \pi\right)^{15/8} + C \).

Step 1: Identify the Substitution

The integral is given as \( \int v\left(\sqrt{3} v^{2}+\pi\right)^{7 / 8} \, dv \). We should look for a substitution that simplifies the expression inside the integral. Notice the expression \( \sqrt{3} v^{2} + \pi \) inside the parentheses. Let's set \( u = \sqrt{3} v^{2} + \pi \).

Step 2: Differentiate the Substitution

Differentiate both sides of the substitution \( u = \sqrt{3} v^{2} + \pi \) with respect to \( v \). This gives \( \frac{du}{dv} = 2\sqrt{3}v \). Thus, \( du = 2\sqrt{3}v \, dv \).

Step 3: Solve for \( dv \)

From \( du = 2\sqrt{3}v \, dv \), solve for \( dv \) to replace it in the integral. We have: \( dv = \frac{du}{2\sqrt{3}v} \).

Step 4: Substitute and Simplify the Integral

Substitute \( u = \sqrt{3} v^{2} + \pi \) and \( dv = \frac{du}{2\sqrt{3}v} \) into the integral. The integral becomes: \( \frac{1}{2\sqrt{3}} \int v u^{7/8} \, \frac{du}{v} \). The \( v \) terms cancel out, simplifying the integral to \( \frac{1}{2\sqrt{3}} \int u^{7/8} \, du \).

Step 5: Integrate with Respect to \( u \)

Now integrate \( \frac{1}{2\sqrt{3}} \int u^{7/8} \, du \). The integral of \( u^{7/8} \) is \( \frac{u^{15/8}}{15/8} \), or equivalently \( \frac{8}{15}u^{15/8} \). Therefore, we have: \( \frac{1}{2\sqrt{3}} \times \frac{8}{15} u^{15/8} + C \).

Step 6: Back Substitute \( u \) to \( v \)

Replace \( u \) back with the original substitution \( u = \sqrt{3} v^{2} + \pi \). The solution is \( \frac{4}{15\sqrt{3}}\left(\sqrt{3} v^{2} + \pi\right)^{15/8} + C \).

Key Concepts

Substitution Method

The substitution method is a crucial tool in calculus for solving indefinite integrals. It's particularly useful when you encounter complex expressions that may seem difficult to integrate directly. The central concept of substitution is to simplify the integral by changing variables.

• First, you identify a part of the integrand (the function being integrated) to substitute with a new variable, typically represented as u. This new variable is chosen because it simplifies the expression.
• Next, you find the derivative of your substitution with respect to the original variable. This step allows you to replace all instances of the original variable and its differential.
• Once substituted, the integral should be simplified so that it becomes easier to solve.

In the original exercise, we use the substitution \( u = \sqrt{3} v^{2} + \pi \). This choice is strategic as it turns a seemingly daunting expression into a more manageable form, making the integration process more straightforward.

Integration Techniques

Integration techniques are essential for solving calculus problems that involve indefinite integrals. Different techniques provide a toolkit for handling various forms and complexities of functions. One frequently used technique is substitution, as demonstrated in the exercise.

• To perform substitution effectively, one must choose a part of the integrand that, when substituted, simplifies the problem.
• After identifying the substitution, reconfigure the integral by expressing the whole function and its differential in terms of the new variable.
• Finally, carry out the integration. Often, this step involves basic integration rules and sometimes a bit of rearranging.

By applying these steps in the exercise, we transitioned from an expression involving \( v \) to one entirely in terms of \( u \), which simplified the calculation to \( \frac{1}{2\sqrt{3}} \int u^{7/8} \, du \). This simplification highlights the power and utility of the substitution method within various integration techniques.

Calculus Problem Solving

The process of solving calculus problems often requires a combination of intuition, strategy, and mathematical tools. Understanding when and how to apply techniques like substitution is pivotal to problem-solving success. Here are a few practical steps you can follow:

• First, analyze the integral to identify if direct integration is possible or whether a technique like substitution is needed.
• Next, think about what transformations can make the problem more tractable. These transformations often involve changing variables or simplifying expressions.
• Once you have solved the integral, ensure to back substitute if needed, bringing the solution back to terms of the original variables.

Always check your work by differentiating your result to see if it matches the original integrand. In our exercise, the process included identifying the substitution \( u = \sqrt{3} v^{2} + \pi \), simplifying the integral, and re-expressing the solution in terms of the original variable \( v \). With practice, these techniques become second nature, making problem solving in calculus more effective and efficient.