Problem 55 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int x(x+1) d x $$

Step-by-Step Solution

Answer

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

Step 1: Expand the Expression

Start by expanding the integrand. Distribute \(x\) across \((x+1)\) to rewrite the expression.\[x(x+1) = x^2 + x.\]The integral becomes:\[\int x(x+1) \, dx = \int (x^2 + x) \, dx.\]This lets us work with simpler polynomial terms.

Step 2: Integrate Each Term Separately

Now, integrate each term in the expanded expression separately. The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), for any constant \(n\).\[\int (x^2 + x) \, dx = \int x^2 \, dx + \int x \, dx.\]Integrate each term to obtain:\[\frac{x^3}{3} + \frac{x^2}{2}.\]

Step 3: Add the Constant of Integration

Indefinite integrals include a constant of integration \(C\) since the antiderivative is determined up to a constant. So, the complete integrated expression becomes:\[\frac{x^3}{3} + \frac{x^2}{2} + C.\]Write the final expression as the solution.

Key Concepts

Integration Techniques

When we talk about indefinite integrals, understanding integration techniques becomes essential. If you're new to integration, you can imagine it as the reverse of differentiation. In differentiation, you find how fast something changes, such as the speed of a car. Integration, though, helps to calculate the accumulation of that change over a time period, like finding the total distance traveled by the car.

For the exercise given, we apply basic polynomial integration. But before integrating, there's a trick called 'expansion' that helps simplify the process. Think of it like trying to describe a word by spelling it out first. Here, we expand the expression \(x(x+1)\) to \(x^2 + x\).

• First, distribute \(x\) across \((x+1)\) to get a clearer form.
• This ensures we handle each part individually with ease.

By breaking down complex expressions, we pave the way for straightforward integration, making the whole process much more manageable.

Polynomial Integration

Polynomial integration is one of the basic techniques used to find antiderivatives. When dealing with polynomials, each term follows a simple power rule: the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). Imagine handling each component separately—they are like different ingredients in a recipe.

Here's how it's done:

• For \(x^2\), using the power rule gives \(\frac{x^3}{3}\).
• And for \(x\), it results in \(\frac{x^2}{2}\).

The beauty of polynomial integration lies in its straightforwardness. Each term is handled independently, ensuring no mix-ups or confusion along the way.

So, when you see a polynomial, remember: just apply the power rule to each term and enjoy your easy-peasy solution. By dissecting the polynomial into manageable chunks, the complicated-looking expression turns into a simple summation of terms.

Constant of Integration

A unique aspect of indefinite integrals is the constant of integration, often denoted as \(C\). When finding antiderivatives, multiple potential functions could fit the bill due to the nature of derivatives.

Why, you ask? Here's why:

• A derivative of a constant is zero.
• So, different constants attached to the same function give the same derivative result.

This makes the constant \(C\) crucial, reminding us that an infinite number of antiderivatives exist. Always include \(C\) in your indefinite integrals.

Take note: leaving out \(C\) would mean ignoring all those possibilities. It's like baking cookies and forgetting to add sugar—you end up with something incomplete. Indicate the broad range of potential solutions with \(C\), ensuring mathematical accuracy and completeness in your learning journey.