If `int(e^(x)(1+100x^(99)-x^(200)))/((1-x^(100))sqrt(1-x^(200)))dx=e^(x)((1-x^(k))/(1+x^(l)))^(m)+c`, then

Options

A. `k = 100, n=1`
B. `l=99, m=(1)/(2)`
C. `k=99, m=1`
D. `k=100, m=-(1)/(2)`

Explanation

To solve the given integral problem, we will break it down step by step.

### Step 1: Understand the Integral
We are given the integral:
\[ I = \int \frac{e^x (1 + 100x^{99} - x^{200})}{(1 - x^{100}) \sqrt{1 - x^{200}}} \, dx \]
and we need to express it in the form:
\[ e^x \left( \frac{1 - x^k}{1 + x^l} \right)^m + C \]
where we need to find the values of \(k\) and \(m\).

### Step 2: Rearranging the Numerator
We can rearrange the numerator:
\[ 1 + 100x^{99} - x^{200} = (1 - x^{200}) + 100x^{99} \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{e^x \left( (1 - x^{200}) + 100x^{99} \right)}{(1 - x^{100}) \sqrt{1 - x^{200}}} \, dx \]

### Step 3: Split the Integral
We can split the integral into two parts:
\[ I = \int \frac{e^x (1 - x^{200})}{(1 - x^{100}) \sqrt{1 - x^{200}}} \, dx + \int \frac{100 e^x x^{99}}{(1 - x^{100}) \sqrt{1 - x^{200}}} \, dx \]

### Step 4: Simplifying the First Integral
For the first integral:
\[ \int \frac{e^x (1 - x^{200})}{(1 - x^{100}) \sqrt{1 - x^{200}}} \, dx \]
we can use the identity \(1 - x^{200} = (1 - x^{100})(1 + x^{100})\):
\[ = \int \frac{e^x (1 - x^{100})(1 + x^{100})}{(1 - x^{100}) \sqrt{1 - x^{200}}} \, dx \]
This simplifies to:
\[ = \int \frac{e^x (1 + x^{100})}{\sqrt{1 - x^{200}}} \, dx \]

### Step 5: Finding the Derivative
Let:
\[ f(x) = \frac{1 + x^{100}}{\sqrt{1 - x^{200}}} \]
We need to find the derivative \(f'(x)\) using the quotient rule.

### Step 6: Applying the Quotient Rule
Using the quotient rule:
\[ f'(x) = \frac{(100x^{99})\sqrt{1 - x^{200}} - (1 + x^{100}) \cdot \frac{-200x^{199}}{2\sqrt{1 - x^{200}}}}{1 - x^{200}} \]
This simplifies to:
\[ f'(x) = \frac{100x^{99}\sqrt{1 - x^{200}} + 100x^{199}(1 + x^{100})}{(1 - x^{200})\sqrt{1 - x^{200}}} \]

### Step 7: Integral Evaluation
Now, we can express the integral \(I\) in terms of \(f(x)\):
\[ I = e^x f(x) + C \]

### Step 8: Expressing in Required Form
We need to express \(f(x)\) in the form \(\frac{1 - x^k}{1 + x^l}\):
\[ f(x) = \frac{1 + x^{100}}{\sqrt{1 - x^{200}}} = e^x \left( \frac{1 - x^{100}}{1 + x^{100}} \right)^{-1/2} \]
From this, we can identify:
- \(k = 100\)
- \(m = -\frac{1}{2}\)

### Final Answer
Thus, the values of \(k\) and \(m\) are:
\[ k = 100, \quad m = -\frac{1}{2} \]