Problem 45 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral is \(\frac{4}{5}x^{5/2} - \frac{2}{3}x^{3/2} + C\).

Step 1: Simplify the Integrand

Start by simplifying the expression inside the integral. The function given is \(\frac{2x^2 - x}{\sqrt{x}}\). We can simplify each term by dividing by \(\sqrt{x}\). Rewrite it as \(2x^2 \cdot x^{-1/2} - x \cdot x^{-1/2}\). This results in \(2x^{3/2} - x^{1/2}\).

Step 2: Integrate Each Term Separately

Integrate each term individually by adding 1 to the power and dividing by the new power. For \(2x^{3/2}\), the integral is \(\frac{2}{5/2}x^{5/2} = \frac{4}{5}x^{5/2}\). For \(x^{1/2}\), the integral is \(\frac{2}{3}x^{3/2}\).

Step 3: Combine the Results

Combine the results from each term's integration and add the constant of integration \(C\). This gives \(\frac{4}{5}x^{5/2} - \frac{2}{3}x^{3/2} + C\).

Key Concepts

Indefinite Integrals

Indefinite integrals, often symbolized by the integral sign (∫) without specified limits, represent the collection of all antiderivatives of a function. This is like reversing the process of differentiation. When we find the indefinite integral of a function, we are essentially looking for a function whose derivative brings us back to the original function.

The basic form is \( \\int\u00a0f(x)\u00a0dx =\u00a0F(x)+C\), where \(F(x)\) is an antiderivative of \(f(x)\), and \(C\) is the constant of integration.

• \(C\) accounts for the indefinite nature since any constant added to a function doesn't change its derivative.
• Unlike definite integrals, indefinite integrals don't have upper and lower bounds.

Thus, indefinite integrals encompass a family of functions and require careful manipulation and understanding of basic functions to solve.

Integration Techniques

Integration techniques are the methods used to solve integrals, which can vary depending on the form of the function being integrated. Here are a few basic and essential techniques that are often utilized:

• Simplification: This is the preliminary step where factors of the function are simplified to make the integration process manageable. This was seen in our exercise where \(2x^2 - x\u00a0\) was simplified to \(2x^{3/2} - x^{1/2}\u00a0\).


• Integration by parts: Often applicable when the function is a product of two different kinds, like polynomials and exponentials.


• Substitution: Useful when the integrand involves a composite function, sometimes requiring a proper replacement of variables to simplify integration.

Having a good grasp of these techniques, as well as practicing their application, is crucial for effectively solving various integrals, as different functions require different approaches.

Power Rule for Integration

The power rule is one of the simplest and most practical tools in integration. It is particularly used when you can express the integrand (the function to be integrated) in terms of power functions. The power rule states that for a function \(x^n\), where \(neq-1\), the integral is:

\[\int x^n dx = \frac{x^{n+1}}{n+1} + C\]

This indicates:

• You add 1 to the exponent \(n\).
• Then, divide the entire term by the new exponent \(n+1\).
• Remember to include \(C\), the constant of integration.

Applying this rule correctly, as demonstrated in our exercise with terms like \(2x^{3/2}\u00a0\) and \(x^{1/2}\u00a0\), simplifies the process significantly and aids in reaching the solution methodically.