Problem 2 - Applied Calculus

Question

Find each indefinite integral. [Integration formulas (A), (B), and (C) are on the inside back cover, numbered \(5-7]\) $$ \int\left(x^{3}+1\right)^{4} 3 x^{2} d x \quad\left[\begin{array}{l} \text { Hint: Use } u=x^{3}+1 \\ \text { and formula } 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{(x^3 + 1)^5}{5} + C \).

Step 1: Identify Substitution Variable

We will perform substitution to simplify the integral. The hint suggests letting \( u = x^3 + 1 \). This substitution will allow us to use a basic integration formula.

Step 2: Find the Differential

Differentiate the substitution variable \( u \) to find \( du \). Since \( u = x^3 + 1 \), we differentiate with respect to \( x \) to get \( \frac{du}{dx} = 3x^2 \). This implies \( du = 3x^2 dx \).

Step 3: Substitute into the Integral

Replace \( x^3 + 1 \) with \( u \) and replace \( 3x^2 dx \) with \( du \) in the integral. The integral becomes \( \int u^4 \, du \).

Step 4: Integrate using Formula

Now that the integral is in a simpler form, apply the power rule for integration (Formula 5), which states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \). Using \( n = 4 \), the integral becomes \( \frac{u^5}{5} + C \).

Step 5: Substitute Back

Substitute back the original variable \( x \) into the integrated function. Since \( u = x^3 + 1 \), the result is \( \frac{(x^3 + 1)^5}{5} + C \).

Key Concepts

Substitution in Integration

The substitution in integration is a powerful technique for simplifying complex integrals. By changing variables, we can transform a difficult integral into one that is easier to evaluate. This process involves choosing a new variable, often denoted as \( u \), to replace a more complicated expression in the integral.

In this exercise, we are asked to find the indefinite integral of \( (x^3 + 1)^4 \, 3x^2 \, dx \). Using the substitution \( u = x^3 + 1 \), we can simplify the expression significantly. When substituting, it is important to also find the differential \( du \) in terms of \( dx \). This involves differentiating the expression for \( u \).

The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 3x^2 \), which implies \( du = 3x^2 \, dx \). Now, you replace the original integral with the new variable and its differential, converting the original expresssion into an integral that is easier to solve.

Integration Formulas

Integration formulas are essential tools that make evaluating integrals more manageable. These formulas provide set patterns or rules to follow when integrating different functions. Many basic integration formulas are based on common function types such as powers, exponentials, and trigonometric functions.

In this exercise, we are specifically using a formula from the integration table: the power rule for integration. This formula is particularly useful for integrating expressions that contain powers of a variable. It provides a straightforward approach that simplifies the integration task.

• Make sure to correctly identify which formula is applicable, which is particularly crucial during the substitution process.
• Accurate use of integration formulas can save significant time and reduce errors.

Understanding these formulas thoroughly helps in quickly selecting and applying the right one to the integral at hand.

Power Rule for Integration

The power rule for integration is one of the most fundamental integration techniques. It allows us to integrate functions of the form \( x^n \) easily. The power rule states that the integral of \( u^n \) with respect to \( u \) is \( \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \). This rule assumes prominence because of its simplicity and wide applicability.

When applying the power rule, it's critical to adjust the exponent and divide by the new exponent. In our case, we have \( \int u^4 \, du \). Using the power rule, it becomes \( \frac{u^5}{5} + C \). This step effectively solves the integral, turning it into a simple algebraic expression.

Finally, once the integration is complete, remember to back-substitute the original variables. This means replacing \( u \) back with \( x^3 + 1 \), leading us to the final result: \( \frac{(x^3 + 1)^5}{5} + C \). This complete cycle from substitution to back-substitution underscores the utility and importance of the power rule.