91Ó°ÊÓ

The binomial theorem is a powerful tool to expand expressions raised to a power. It states that for any numbers \(a\) and \(b\), and a non-negative integer \(n\): \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \].
In this exercise, we used the binomial theorem to expand \((2x+1)^3\). By identifying \(a = 2x\), \(b = 1\), and \(n = 3\), we can expand:
\[(2x+1)^3 = \binom{3}{0}(2x)^3(1)^0 + \binom{3}{1}(2x)^2(1)^1 + \binom{3}{2}(2x)^1(1)^2 + \binom{3}{3}(2x)^0(1)^3 \].
The binomial coefficients \(\binom{3}{k}\) are derived from Pascal's Triangle. This expands to: \[ (2x+1)^3 = 8x^3 + 12x^2 + 6x + 1 \]. By expanding, complex expressions are simplified into a summation of polynomial terms, making integration easier.

Polynomial expansion involves expressing a polynomial, like \((2x + 1)^3\), as a sum of simpler polynomial terms. In our exercise, we used the binomial theorem to rewrite \((2x + 1)^3\) into individual polynomial terms: \[ (2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1 \].
Each term of the expanded polynomial can then be integrated separately:

• \int 8x^3 \, dx = 2x^4 + C_1 \
• \int 12x^2 \, dx = 4x^3 + C_2 \
• \int 6x \, dx = 3x^2 + C_3 \
• \int 1 \, dx = x + C_4 \

The constants of integration combine into one, giving us the result:
\[ 2x^4 + 4x^3 + 3x^2 + x + C \].
This method helps in visualizing and solving integrals efficiently.

The substitution method simplifies integration by substituting a part of the integral with a new variable. For \((2x + 1)^3\), letting \(u = 2x + 1\) transforms the integral.
By differentiating, we get \(du = 2 \, dx\) or \(dx = \frac{du}{2}\). This changes the original integral into a simpler form: \[\int (2x+1)^3 \, dx = \int u^3 \, \frac{du}{2} = \frac{1}{2} \int u^3 \, du \].
We perform the integration:
\[ \frac{1}{2} \left( \frac{u^4}{4} \right) + C = \frac{u^4}{8} + C \].
Finally, substituting \(u\) back, we obtain: \[ \frac{(2x+1)^4}{8} + C \].
This method is useful for integrals with compositions of functions, allowing for easier computation by changing variables.

Definite integrals calculate the exact area under a curve between two bounds. Unlike indefinite integrals, definite integrals evaluate to a specific number.
Suppose we want to evaluate \int_{a}^{b} (2x+1)^3 \, dx\. Using substitution, let \(u = 2x + 1\), hence the bounds change accordingly.
For \int_{a}^{b} (2x+1)^3 \, dx\ with \( u = 2x + 1 \):

• \int_{a}^{b} f(x) \, dx = \int_{u(a)}^{u(b)} f(g(u)) \, du \

This ensures the values respect the transformed variable.
Exact bounds must be calculated from the original limits.
Definite integrals can often be simplified using properties of antiderivatives, making it straightforward to compute areas and physical quantities under a curve.