To solve the problem, we need to find the value of \(\lambda\) such that \(I_1 = \lambda I_2\), where
\[
I_1 = \int_0^1 (1 - x^{50})^{100} \, dx
\]
and
\[
I_2 = \int_0^1 (1 - x^{50})^{101} \, dx.
\]
### Step 1: Express \(I_2\) in terms of \(I_1\)
We can rewrite \(I_2\) as follows:
\[
I_2 = \int_0^1 (1 - x^{50})^{101} \, dx = \int_0^1 (1 - x^{50})^{100} (1 - x^{50}) \, dx.
\]
### Step 2: Expand \(I_2\)
Now we can expand \(I_2\):
\[
I_2 = \int_0^1 (1 - x^{50})^{100} \, dx - \int_0^1 x^{50} (1 - x^{50})^{100} \, dx.
\]
### Step 3: Identify \(I_1\)
From the definition of \(I_1\), we have:
\[
I_1 = \int_0^1 (1 - x^{50})^{100} \, dx.
\]
### Step 4: Define the second integral
Let us denote the second integral as \(I\):
\[
I = \int_0^1 x^{50} (1 - x^{50})^{100} \, dx.
\]
### Step 5: Relate \(I_1\) and \(I_2\)
Now we can express \(I_2\) in terms of \(I_1\) and \(I\):
\[
I_2 = I_1 - I.
\]
### Step 6: Find \(I\)
To find \(I\), we can use integration by parts. Let:
- \(u = (1 - x^{50})^{100}\) and \(dv = x^{50} \, dx\).
Then:
- \(du = -50 x^{49} (1 - x^{50})^{99} \, dx\) and \(v = \frac{x^{51}}{51}\).
Using integration by parts:
\[
I = \left[ u v \right]_0^1 - \int_0^1 v \, du.
\]
Evaluating the boundary term:
At \(x = 1\), \(u = 0\) and at \(x = 0\), \(u = 1\), so:
\[
\left[ u v \right]_0^1 = 0 - 0 = 0.
\]
Now we need to evaluate the integral:
\[
-\int_0^1 \frac{x^{51}}{51} \left(-50 x^{49} (1 - x^{50})^{99}\right) \, dx = \frac{50}{51} \int_0^1 x^{100} (1 - x^{50})^{99} \, dx.
\]
### Step 7: Substitute back into \(I_2\)
Now we can substitute \(I\) back into the equation for \(I_2\):
\[
I_2 = I_1 - \frac{50}{51} I_2.
\]
### Step 8: Solve for \(I_2\)
Rearranging gives:
\[
I_2 + \frac{50}{51} I_2 = I_1,
\]
which simplifies to:
\[
\frac{101}{51} I_2 = I_1.
\]
### Step 9: Find \(\lambda\)
Thus, we find:
\[
\lambda = \frac{I_1}{I_2} = \frac{101}{51}.
\]
### Final Answer
The value of \(\lambda\) is:
\[
\lambda = \frac{101}{51}.
\]
