`int e^x (1 + 10 x^9 - x^20)/((1 - x^(10))sqrt(1 - x^(20))) dx` equal to: (where c is an integral constant)

Options

A. `e^(x)((1 - x^(10))/(1 + x^(10))) + c`
B. `e^(x)((1 + x^(10))/(1 - x^(10))) + c`
C. `e^(x)(sqrt((1 + x^(10))/(1 - x^(10)))) + c`
D. `e^(x)(sqrt((1 - x^(10))/(1 + x^(10)))) + c`

Explanation

To solve the integral

\[ \int e^x \frac{1 + 10 x^9 - x^{20}}{(1 - x^{10})\sqrt{1 - x^{20}}} \, dx, \]

we can break it down into simpler parts.

### Step 1: Rewrite the Integral
We can split the integral into two parts:

\[ \int e^x \left( \frac{1 - x^{20}}{(1 - x^{10})\sqrt{1 - x^{20}}} + \frac{10 x^9}{(1 - x^{10})\sqrt{1 - x^{20}}} \right) \, dx. \]

### Step 2: Simplify the First Part
For the first part, we can simplify:

\[ \frac{1 - x^{20}}{(1 - x^{10})\sqrt{1 - x^{20}}} = \frac{(1 - x^{10})(1 + x^{10})}{(1 - x^{10})\sqrt{1 - x^{20}}} = \frac{1 + x^{10}}{\sqrt{1 - x^{20}}}. \]

Thus, the integral becomes:

\[ \int e^x \left( \frac{1 + x^{10}}{\sqrt{1 - x^{20}}} + \frac{10 x^9}{(1 - x^{10})\sqrt{1 - x^{20}}} \right) \, dx. \]

### Step 3: Identify the Function \( f(x) \)
Let

\[ f(x) = \frac{1 + x^{10}}{\sqrt{1 - x^{20}}}. \]

### Step 4: Differentiate \( f(x) \)
We need to find \( f'(x) \) to use the integration by parts formula:

\[ \int e^x f'(x) \, dx = e^x f(x) + C. \]

Using the quotient and chain rule, we differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} \left( \frac{1 + x^{10}}{\sqrt{1 - x^{20}}} \right). \]

This will involve applying the product and chain rules.

### Step 5: Solve the Integral
After finding \( f'(x) \), we can substitute back into the integral:

\[ \int e^x f'(x) \, dx = e^x f(x) + C. \]

### Step 6: Final Expression
Thus, the final result of the integral is:

\[ e^x \left( \frac{1 + x^{10}}{\sqrt{1 - x^{20}}} \right) + C. \]