If `(d)/(dx)[ x^(n+1)+c]=(n+1)x^(n)`, then find `int x^(n)dx`.

Answer

To solve the problem, we start with the given differentiation:

\[ \frac{d}{dx}[x^{n+1} + c] = (n+1)x^n \]

We want to find the integral:

\[ \int x^n \, dx \]

### Step-by-Step Solution:

1. **Integrate Both Sides**:
We will integrate both sides of the given equation with respect to \(x\).

\[ \int \frac{d}{dx}[x^{n+1} + c] \, dx = \int (n+1)x^n \, dx \]

2. **Apply the Fundamental Theorem of Calculus**:
The left side simplifies to:

\[ x^{n+1} + c \]

The right side can be integrated as follows:

\[ \int (n+1)x^n \, dx = (n+1) \int x^n \, dx \]

3. **Rearranging the Equation**:
Now we have:

\[ x^{n+1} + c = (n+1) \int x^n \, dx \]

4. **Isolate the Integral**:
To find \(\int x^n \, dx\), we rearrange the equation:

\[ \int x^n \, dx = \frac{x^{n+1} + c}{n+1} \]

5. **Final Result**:
We can express the constant \(c\) as a new constant \(C\) (the integration constant):

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

### Final Answer:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]