Results Summary
If you’re searching for “integration of x 2 2,” here are some possible interpretations and their solutions:
$$\boxed{\int \frac{x – 2}{2} \, dx = \frac{x^2}{4} – x + C}$$
$$\boxed{\int (x – 2)^2 \, dx = \frac{x^3}{3} – 2x^2 + 4x + C}$$
$$\boxed{\int \frac{x^2}{2} \, dx = \frac{x^3}{6} + C}$$
$$\boxed{\int \frac{x}{2} + 2 \, dx = \frac{x^2}{4} + 2x + C}$$
Introduction
The “integration of x 2 2” could have multiple meanings. Here are four possible options with detailed solutions for each.
Option 1: \( \displaystyle \int \frac{x – 2}{2} \, dx \)
Let’s separate the terms:
\[\begin{align*} = \frac{1}{2} \int x \, dx – \frac{1}{2} \int 2 \, dx \end{align*}\]
Now integrate each term:
\[\begin{align*} = \frac{1}{2} \cdot \frac{x^2}{2} – \frac{1}{2} \cdot 2x + C \end{align*}\]
Simplify:
\[\begin{align*} = \frac{x^2}{4} – x + C \end{align*}\]
Thus, the result is:
\[\begin{align*} \int \frac{x – 2}{2} \, dx = \frac{x^2}{4} – x + C \end{align*}\]
Option 2: \( \displaystyle \int (x – 2)^2 \, dx \)
Let’s expand \( (x – 2)^2 \):
\[\begin{align*} = \int (x^2 – 4x + 4) \, dx \end{align*}\]
Now integrate each term:
\[\begin{align*} = \frac{x^3}{3} – 2x^2 + 4x + C \end{align*}\]
Thus, the result is:
\[\begin{align*} \int (x – 2)^2 \, dx = \frac{x^3}{3} – 2x^2 + 4x + C \end{align*}\]
Option 3: \( \displaystyle \int \frac{x^2}{2} \, dx \)
Here, we factor out the constant \( \frac{1}{2} \):
\[\begin{align*} = \frac{1}{2} \int x^2 \, dx \end{align*}\]
Using the power rule:
\[\begin{align*} = \frac{1}{2} \cdot \frac{x^3}{3} + C = \frac{x^3}{6} + C \end{align*}\]
Thus, the result is:
\[\begin{align*} \int \frac{x^2}{2} \, dx = \frac{x^3}{6} + C \end{align*}\]
Option 4: \( \displaystyle \int \frac{x}{2} + 2 \, dx \)
We first separate the terms:
\[\begin{align*} = \int \frac{x}{2} \, dx + \int 2 \, dx \end{align*}\]
Now integrate each term:
\[\begin{align*} = \frac{x^2}{4} + 2x + C \end{align*}\]
Thus, the result is:
\[\begin{align*} \int \frac{x}{2} + 2 \, dx = \frac{x^2}{4} + 2x + C \end{align*}\]