Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

nd the inde nite integrals. (a) \(\int\left(x^{2}+3\right)^{3} x d x\) (b) \(\int x \sqrt{x^{2}+4} d x\) (c) \(\int(2 x+1) \sqrt{x^{2}+x} d x\)

Step-by-Step Solution

Answer

The indefinite integrals are: (a) \( \frac{1}{8}(x^{2}+3)^{4} + C \), (b) \( \frac{1}{3}(x^{2}+4)^{\frac{3}{2}} + C \), and (c) \( \frac{2}{3}(x^{2}+x)^{\frac{3}{2}} + C \).

Step 1: Integral of (x^2 + 3)^3 * x dx

Let \( u = x^{2}+3 \), then the derivative \( du = 2x \, dx \). The integral then becomes: \( \frac{1}{2} \int u^{3} du \). Apply the power rule for integrals to get \( \frac{1}{8}u^{4} + C \) where C is the constant of integration. Substituting back \( u = x^{2}+3 \), the final result is \( \frac{1}{8}(x^{2}+3)^{4} + C \)

Step 2: Integral of x sqrt(x^2 + 4) dx

Let \( u = x^{2}+4 \), then the derivative \( du = 2x \, dx \). The integral then becomes: \( \frac{1}{2} \int \sqrt{u} du \). Use the power rule for integrals to get \( \frac{1}{3}u^{\frac{3}{2}} + C \). Substituting back \( u = x^{2}+4 \), the final result is \( \frac{1}{3}(x^{2}+4)^{\frac{3}{2}} + C \)

Step 3: Integral of (2x + 1) sqrt(x^2 + x) dx

Let \( u = x^{2}+x \), then the derivative \( du = (2x + 1)dx \). The integral then becomes: \( \int u^{\frac{1}{2}} du \). Apply the power rule for integrals to get \( \frac{2}{3}u^{\frac{3}{2}}+ C \). Substituting back \( u = x^{2}+x \), the final result is \( \frac{2}{3}(x^{2}+x)^{\frac{3}{2}} + C \)

Key Concepts

Substitution Method

The substitution method is a popular integration technique used to simplify complex integrals. It works by replacing a complicated part of the integral with a single variable, often transforming it into a simpler form. The key idea is to recognize a portion of the integrand that, when replaced, makes the integral easier to evaluate. This is often paired with changing the differential accordingly.
\[ \begin{align*} \text{For example, consider the integral:} \quad &\int (x^2+3)^3 x \, dx. \ \text{First, we set:} \quad &u = x^2 + 3. \ \text{Then, its differential is:} \quad &du = 2x \, dx. \ \text{Therefore,} \quad &x \, dx = \frac{1}{2} du. \end{align*} \] This substitution transforms the original integral into \( \frac{1}{2} \int u^3 \, du \), which is much simpler to handle.
By applying substitution, you replace a complex segment and get a cleaner formulation, making the integration process much easier.

Power Rule for Integration

The power rule for integration is a fundamental tool in calculus used to integrate functions of the form \( x^n \). It provides the antiderivative for these polynomial-type expressions, and it is as straightforward as adding one to the power of \( x \) and dividing by the new exponent.
Here's the power rule formula used in our examples:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C. \]
Where \( C \) is the constant of integration, emphasized because the indefinite integral is representing a family of functions rather than a specific solution.
In the integral \( \frac{1}{2} \int u^3 \, du \), we apply the power rule:
\[ \frac{1}{2} \times \frac{u^{4}}{4} = \frac{1}{8}u^{4} \] plus the constant of integration \( C \). This transforms our integrated function back after substituting the initial expression for \( u \). It’s essential to practice this rule, as it is foundational for handling polynomial integrals effectively.

Integration Techniques

Integration techniques are methods used to evaluate integrals. They provide strategies for tackling a wide range of integrand forms that are not readily integrable by basic rules.
In our examples, we utilized two primary techniques:

• **Substitution Method:** It's effective for integrals where the integrand can be expressed as a composite function or when one part of the function's derivative is present in the integral.
• **Power Rule for Integration:** It’s used for polynomial functions, where each term can straightforwardly be integrated separately by increasing the exponent and dividing by the new power, as seen in \( \int u^3 \, du \).

When tackling more complex integrals, it's crucial to recognize patterns and decide on the appropriate integration technique. Mastery of these techniques elevates your ability to handle seemingly intricate problems with confidence, ensuring a smoother problem-solving experience.