Problem 48 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(1+x^{3}\right) \sqrt{x} d x $$

Step-by-Step Solution

Answer

\( \frac{2}{3} x^{1.5} + \frac{2}{9} x^{4.5} + C \)

Step 1: Identify the Integral Structure

The integral we need to solve is \( \int \left(1 + x^3\right) \sqrt{x} \; dx \). This expression can be expanded by distributing \( \sqrt{x} \) inside the parenthesis to simplify the integration process.

Step 2: Expand the Integrand

Distribute \( \sqrt{x} \) into the terms inside the parenthesis:\[\int \left(1 \cdot \sqrt{x} + x^3 \cdot \sqrt{x}\right) \, dx = \int \left(\sqrt{x} + x^{3.5}\right) \, dx\]This simplifies the expression to two separate integrals.

Step 3: Rewrite Exponents as Fractions

Since \( \sqrt{x} = x^{0.5} \), rewrite the integrals using fractional exponents:\[\int \left(x^{0.5} + x^{3.5}\right) \, dx\]This format is easier to integrate term by term.

Step 4: Apply the Power Rule

Apply the power rule for integration, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), to each term individually:\[\int x^{0.5} \, dx = \frac{x^{0.5+1}}{0.5+1} = \frac{x^{1.5}}{1.5}\]\[\int x^{3.5} \, dx = \frac{x^{3.5+1}}{3.5+1} = \frac{x^{4.5}}{4.5}\]

Step 5: Simplify the Integrated Terms

Combine the integrated results:\[\int \left(x^{0.5} + x^{3.5}\right) \, dx = \frac{x^{1.5}}{1.5} + \frac{x^{4.5}}{4.5}\]Rewriting the fractions, you get:\[\frac{2}{3}x^{1.5} + \frac{2}{9}x^{4.5}\]

Step 6: Combine and Add the Constant of Integration

Add the constant of integration \( C \), since this is an indefinite integral:\[\frac{2}{3} x^{1.5} + \frac{2}{9} x^{4.5} + C\]

Step 7: Final Form of the Integral

Subtracting constants is not needed here. The solution to the integral is:\[\frac{2}{3} x^{1.5} + \frac{2}{9} x^{4.5} + C\]

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that makes integrating expressions with exponents straightforward and manageable. It is primarily used for finding the indefinite integral of terms of the form \(x^n\). The rule is quite simple and can be expressed as follows:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \(n\) is a real number, and \(C\) is the constant of integration, which is essential in indefinite integrals as it accounts for the family of antiderivatives.

The power rule simplifies the process by allowing us to deal with exponents directly without rewriting parts of the integroal notes. This makes it incredibly handy when you're expanding expressions like \(1 + x^3\) and dealing with terms like \(\sqrt{x}\), which can be rewritten using exponents.

Let's consider an example similar to our problem \( \int (1 + x^3) \sqrt{x} \, dx \). By distributing \(\sqrt{x}\) and rewriting \(\sqrt{x}\) as \(x^{0.5}\), we apply the power rule to each term:

• For \(\int x^{0.5} \, dx = \frac{x^{1.5}}{1.5} + C\), simply increase the exponent by 1 and divide by the new exponent.
• For \(\int x^{3.5} \, dx = \frac{x^{4.5}}{4.5} + C\), it's exactly the same process. Notice how seamless it is to switch between integrals using the power rule.

In the end, this rule saves us time and makes integrating much more intuitive.

Fractional Exponents

Fractional exponents can be a little tricky at first, but they are crucial for rewriting unconventional root expressions into a more manageable exponent form, especially when dealing with integrals. Let's break down what fractional exponents are and how they work.

A fractional exponent like \(x^{\frac{m}{n}}\) is equivalent to \(\sqrt[n]{x^m}\). For example:

• \(\sqrt{x} = x^{0.5}\), which is essentially \(x^{1/2}\).
• \(x^{3.5} = x^{7/2}\), meaning \(x^{(3+0.5)}\) which combines the power operations into one seamless step.

Why is this important?

Working with fractional exponents allows us to efficiently apply rules and patterns, like the power rule in integration, without getting tangled up with roots. In integration problems, simplifying radicals by converting them into fractional exponents keeps everything consistent and linear in structure. For instance, in our equation:
\[\int \left(x^{0.5} + x^{3.5}\right) \, dx\]
By rewriting the original expression \((1 + x^3)\sqrt{x}\), we can break it into recognizable functions, simplifying our integrations significantly. Instead of dealing with roots directly, fractional exponents become numeric, making the whole integration process smooth and methodical.

Integral Calculus

Integral calculus is a core component of calculus concerned with accumulation and area, among other exciting applications. It focuses on finding anti-derivatives and representing them as integrals, like the indefinite integral encountered in this exercise.

An indefinite integral, which is what we are dealing with, does not have set limits of integration. Its purpose is to find a family of functions whose derivative is the integrand. That's why the result always includes a constant \(C\) since multiple functions could exist with the same derivative but differ by a constant.

In the example \(\int(1 + x^3)\sqrt{x} \, dx\), we're calculating an indefinite integral to understand the antiderivative of the expression. Why?

• First, because indefinite integrals help us determine the original function before differentiation.
• Second, they are essential when solving real-world problems involving rates of change, areas under curves, and more.

Integral calculus opens the door to solving a wide range of problems in physics, engineering, economics, and beyond, by allowing us to reverse engineer from rates of change back to the state of functions. The process of finding and adding the constant \(C\) signifies endless potential combinations of solutions within specific parameters.

So, next time you calculate an indefinite integral, think of it as discovering the hidden original pattern behind the presented change.