Problem 46 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{x^{3}+3 x}{2 \sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{7} x^{\frac{7}{2}} + x^{\frac{3}{2}} + C \).

Step 1: Simplify the Expression

First, we simplify the expression inside the integral. Notice that you can break the numerator into two separate terms:\[ \int \frac{x^3}{2\sqrt{x}} \, dx + \int \frac{3x}{2\sqrt{x}} \, dx. \] This can be further simplified into two integrals with each term divided by \( 2\sqrt{x} \).

Step 2: Convert to Power Rule Form

Transform each term to make integration possible. For the first term, \( \frac{x^3}{2\sqrt{x}} \), rewrite it in terms of exponents:\[ = \frac{1}{2} \int x^{3 - \frac{1}{2}} \, dx = \frac{1}{2} \int x^{\frac{5}{2}} \, dx. \]For the second term, \( \frac{3x}{2\sqrt{x}} \), rewrite it as:\[ = \frac{3}{2} \int x^{1 - \frac{1}{2}} \, dx = \frac{3}{2} \int x^{\frac{1}{2}} \, dx. \]

Step 3: Integrate Each Term Independently

Now, integrate each term separately using the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).For \( \frac{1}{2} \int x^{\frac{5}{2}} \, dx \):\[ \frac{1}{2} \cdot \frac{x^{\frac{7}{2}}}{\frac{7}{2}} = \frac{1}{2} \cdot \frac{2}{7} x^{\frac{7}{2}} = \frac{1}{7} x^{\frac{7}{2}}. \]For \( \frac{3}{2} \int x^{\frac{1}{2}} \, dx \):\[ \frac{3}{2} \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = x^{\frac{3}{2}}. \]

Step 4: Final Integration Result

Combine both integrated results and add a constant of integration \( + C \):\[ \frac{1}{7} x^{\frac{7}{2}} + x^{\frac{3}{2}} + C. \]This represents the indefinite integral of the given function.

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus used for integrating powers of variables. It states that to integrate a function of the form \( x^n \), you increase the exponent by one and then divide by the new exponent.
In mathematical terms, the rule is expressed as:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Where: - \( n \) is any real number except -1 - \( C \) is the constant of integration, accounting for any constant term that may have been present before 'integration'.
The power rule simplifies the process of finding antiderivatives and is especially handy when dealing with polynomial functions. In our exercise, each term is rewritten into a power form, allowing the power rule to be directly applied. This step is crucial to convert complex expressions into simpler, integrable forms. By employing the power rule, we efficiently solve for the antiderivatives of each term.

Simplifying Expressions

Before applying the power rule, it's often necessary to simplify the expressions involved in the integral. Simplification helps us transform the integral into a more manageable form.
The original problem involves an expression with a numerator and denominator:

• \( \frac{x^3 + 3x}{2\sqrt{x}} \)

By breaking this expression into two separate fractions, we get:

• \( \frac{x^3}{2\sqrt{x}} + \frac{3x}{2\sqrt{x}} \)

Each term can be further rewritten with exponent notation:

• \( x^{3/2} \) and \( x^{1/2} \)

These transformations not only make the terms ready for the power rule but also highlight a frequent technique in calculus: rewriting radicals and products as power terms. This simplicity in organization paves the way for effective integration, demonstrating why simplification is key in calculus.

Calculus for Biology

Calculus isn't just for physics or pure mathematics—it's incredibly valuable in biology as well. Integrals, like the one we're working with, can model various biological processes.
In the context of biology:

• The power rule can be used to calculate rates of change, such as growth rates of populations.
• Indefinite integrals model cumulative effects, like accumulated concentrations of substances in tissues over time.

The simplification of expressions before integration reflects the need to break down complex biological data into understandable trends and behaviors. Calculus provides critical insights in genomics, pharmacokinetics, and ecological modeling. Understanding these mathematical foundations allows biologists to analyze and predict outcomes in varying biological systems effectively. Thus, mastering these concepts is not only fundamental for mathematicians but also for biologists aiming to apply quantitative analysis in their research efforts.