Problem 66 - Calculus Early Transcendentals

Question

Use a change of variables to evaluate the following integrals. $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the function \( \int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x \) Answer: \( \frac{1}{9} (x^{\frac{3}{2}}+8)^{6} + C \)

Step 1: Choose the substitution variable u

Let's consider the substitution variable u to be the expression inside the parentheses. That is: $$u = x^{\frac{3}{2}} + 8$$

Step 2: Find du/dx

Now, we need to find the derivative of u with respect to x. Using the power rule, we get: $$\frac{du}{dx} = \frac{3}{2} x^{\frac{1}{2}}$$

Step 3: Calculate dx in terms of du

We need to rewrite dx in terms of du, so: $$dx = \frac{du}{\frac{3}{2} x^{\frac{1}{2}}}$$

Step 4: Rewrite the integral using the substitution

We can now rewrite the integral using u and du: $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x = \int u^{5} \sqrt{x} \cdot \frac{du}{\frac{3}{2} x^{\frac{1}{2}}}$$

Step 5: Simplify the integral

Notice that \(\sqrt{x}\) in the numerator and \(x^{\frac{1}{2}}\) in the denominator cancel each other out. We are left with: $$\int u^{5} \cdot \frac{du}{\frac{3}{2}} = \frac{2}{3} \int u^{5} du$$

Step 6: Perform the integration

Now, we can simply evaluate the remaining integral: $$\frac{2}{3} \int u^{5} du = \frac{2}{3} \cdot \frac{u^{6}}{6} + C$$

Step 7: Substitute back for x

Now that we've completed the integration, we need to substitute back for x by replacing u with our original expression: $$\frac{2}{3} \cdot \frac{u^{6}}{6} + C = \frac{2}{18} (x^{\frac{3}{2}} + 8)^{6} + C$$

Step 8: Simplify the final expression

Finally, we simplify the expression to get our final answer: $$\frac{1}{9} (x^{\frac{3}{2}}+8)^{6} + C$$ So, the evaluated integral is: $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x = \frac{1}{9} (x^{\frac{3}{2}}+8)^{6} + C$$

Key Concepts

Change of Variables

When solving integrals, especially complex ones, the change of variables is a powerful technique. Often called "substitution," it involves replacing a variable or expression with a new one. This simplifies the integration process, making it manageable.

• Identify the part to replace: Choose an expression within the integral that complicates it. Generally, this is a nested function or a combination that looks difficult to handle directly. In the given exercise, the complicated expression \( x^{\frac{3}{2}} + 8 \) was chosen for substitution.
• Substitute with a new variable: Assign a new variable \( u \) to this complex expression. This converts the integral into a function of \( u \) instead of \( x \), transforming it to simpler terms.
• Calculate \( du/dx \): Differentiate the new variable with respect to \( x \). This step is crucial as it helps reframe the differential \( dx \) in terms of \( du \), linking \( u \) and \( x \).
• Rewrite the integral: Once the substitution and differentiation are complete, replace \( x \) and \( dx \) in the original integral with \( u \) and \( du \) accordingly.

By easing the integration into simpler terms, this method allows for easier calculation and often leads to cleaner, more intuitive solutions.

Definite and Indefinite Integrals

In calculus, differentiation and integration are inverse operations. When dealing with integrals, we encounter two main types: definite and indefinite integrals. Indefinite integrals, as used in the exercise, express the family of functions. They include a constant \( C \) because integration is the reverse of differentiation. Without boundary values, there's an infinite number of potential solutions shifted by a constant value.

• Symbol: Indefinite integrals are represented by the integral sign \( \int \) followed by the function and \( dx \).
• Result format: The result of indefinite integrals is a function \( F(x) + C \). Here, "\( C \)" stands as a constant of integration.

In contrast, definite integrals provide numerical values, not functions. They involve integrating between specified bounds, giving the net area under a curve between two points.

• Symbol: Written with limits of integration, like \( \int_{a}^{b} \).
• Calculates: Produces a specific number representing area or accumulated quantity.

For the solved exercise, the analysis was an indefinite integral. The substitution ended with a function plus the constant \( C \), typical for indefinite results.

Power Rule in Integration

The power rule is one of the most fundamental tools in integration calculus. It deals with integrating powers of the variable within a function. Given its simplicity and broad applicability, it's indispensable for students learning basics of integration. In the power rule:

• The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \).
• This "plus one, divide by new power" approach is both intuitive and efficient.

In our example, the power rule surfaces when integrating after substitution. Once you rewrite the initial integral using \( u \), the main complexity reduces to integrating \( u^5 \).

• Here, increasing the exponent (\( 5 + 1 \)) gives the result in terms of \( u^{6} \).
• Then, dividing by the new power (6) gives \( \frac{u^6}{6} \).

The beauty of the power rule lies in its straightforward approach, allowing seamless integration of polynomials and similar functions. It is intuitive yet expands with practice, forming the backbone of more advanced topics in calculus.