$$ \int x ^ { n } d x = \frac { x ^ { n + 1 } } { n + 1 } + 0 $$

Question

UpStudy AI · Accepted Answer

The expression you've provided is the formula for the indefinite integral of $$ x^n $$ with respect to $$ x $$, where $$ n \neq -1 $$.

Let's break it down step by step:

1. **Understanding the Integral**: The integral $$ \int x^n \, dx $$ represents the area under the curve of the function $$ x^n $$ with respect to $$ x $$.

2. **Applying the Power Rule**: The power rule for integration states that:
   $$
   \int x^n \, dx = \frac{x^{n+1}}{n+1} + C
   $$
   where $$ C $$ is the constant of integration. This constant is important because the integral represents a family of functions that differ by a constant.

3. **Special Case**: The formula is valid for all $$ n $$ except $$ n = -1 $$. If $$ n = -1 $$, the integral becomes:
   $$
   \int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln |x| + C
   $$

4. **Final Expression**: Therefore, the complete expression for the indefinite integral of $$ x^n $$ is:
   $$
   \int x^n \, dx = \begin{cases} 
   \frac{x^{n+1}}{n+1} + C & \text{if } n \neq -1 \
   \ln |x| + C & \text{if } n = -1 
   \end{cases}
   $$

If you have any specific questions or need further clarification on this topic, feel free to ask!
The integral of $$ x^n $$ with respect to $$ x $$ is $$ \frac{x^{n+1}}{n+1} + C $$, where $$ C $$ is the constant of integration.

\( \int x ^ { n } d x = \frac { x ^ { n + 1 } } { n + 1 } + 0 \)

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Calculus Questions