`int(10 x^9+10 ^xlog_(e^(10))dx)/(x^(10)+10^x)`equals

Options

A. `10^x-x^(10)+C`
B. `10^x+x^(10)+C`
C. `(10^x-x^(10))^(-1)+C`
D. `log(10^x+x^(10))+C`

Explanation

To solve the integral

\[ \int \frac{10x^9 + 10^x \log_{e}(10)}{x^{10} + 10^x} \, dx, \]

we will use substitution and properties of logarithms.

### Step-by-Step Solution:

1. **Identify the substitution**:
Let \( t = x^{10} + 10^x \).

2. **Differentiate the substitution**:
We need to find \( dt \):
\[ dt = \frac{d}{dx}(x^{10} + 10^x) = 10x^9 + 10^x \log_{e}(10) \, dx. \]

3. **Rearranging for \( dx \)**:
From the expression for \( dt \), we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{10x^9 + 10^x \log_{e}(10)}. \]

4. **Substituting in the integral**:
Now substitute \( t \) and \( dx \) into the integral:
\[ \int \frac{10x^9 + 10^x \log_{e}(10)}{x^{10} + 10^x} \, dx = \int \frac{dt}{t}. \]

5. **Integrate**:
The integral of \( \frac{1}{t} \) is:
\[ \int \frac{dt}{t} = \log |t| + C. \]

6. **Back-substitute for \( t \)**:
Replace \( t \) back with \( x^{10} + 10^x \):
\[ \log |x^{10} + 10^x| + C. \]

### Final Answer:
Thus, the solution to the integral is:
\[ \log |x^{10} + 10^x| + C. \]