To fully grasp the concept of integrating polynomials, it is beneficial to first review the process of differentiating them. As integration is the inverse of differentiation, let's start by looking at differentiation from first principles.

From the definition of first principles, we can expand (x+h)^n using a binomial expansion, allowing us to handle each term individually.

When we substitute this into the differentiation formula, we get:

\frac{(x+h)^n - x^n}{h} = \frac{nx^{n-1}h + ...}{h}

By analyzing this term by term, we can see that as h approaches 0, only the first term will remain, as all other terms will contain h.

From this observation, we arrive at the well-known "power rule" for differentiation, which states:

\frac{d}{dx} x^n = nx^{n-1}

With this rule, we can easily integrate polynomials. For instance, if we wanted to integrate x^2 with respect to x:

\int x^2 dx = \frac{x^3}{3} + C

We can also use the linearity property of integrals to break down more complex polynomials into simpler ones. For example, to find:

\int (x^4 + 2x) dx = \int x^4 dx + \int 2x dx = \frac{x^5}{5} + x^2 + C

Let's examine a few additional examples of integrating polynomials.

- Example 1: Find \int x^3 dx.

Using the formula for integrating polynomials, we get:

\int x^3 dx = \frac{x^4}{4} + C

- Example 2: Find \int (x^4 + 3x^2) dx.

We can split this integral into two separate ones using linearity, giving us:

\int (x^4 + 3x^2) dx = \int x^4 dx + \int 3x^2 dx = \frac{x^5}{5} + x^3 + C

- Example 3: Find \int (x-1)^2 (x+1) dx.

To simplify this integral, we can expand the brackets using the binomial theorem, resulting in:

\int (x-1)^2 (x+1) dx = \int (x^3 - 3x^2 + 3x - 1 + 2x^2 - 4x + 2) dx

After using linearity and simplifying, we get:

\int (x^3 - x^2 - x + 1) dx = \frac{x^4}{4} - \frac{x^3}{3} - \frac{x^2}{2} + x + C

We can also integrate this polynomial by using the substitution u = x-1, giving us:

\int (x-1)^2 (x+1) dx = \int (u^2) dx = \frac{u^3}{3} + C = \frac{(x-1)^3}{3} + C

Both methods produce the same result, giving us confidence in our integration.

To recap, here are some crucial things to keep in mind when integrating polynomials:

- The formula for integrating x^n is \int x^n dx = \frac{x^{n+1}}{n+1} + C.
- Integrals have the property of linearity, meaning \int (af + bg) dx = a\int f dx + b\int g dx.
- When dealing with definite integrals, it is vital to consider the limits on each separate integral.

The same formula for integrating x^n applies even when n is a fraction. You can follow the same process as before.

While integrating polynomials may seem intimidating at first, the examples above demonstrate that it can be broken down into simple steps. Remember to utilize the formula for integrating polynomials and the linearity property of integrals to make the process more manageable. With practice, integrating polynomials will become second nature.

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