If `int[sin^2x]/[1+sin^2x] dx` =`x-ktan^-1(M tanx)` then: a. `M=1/sqrt2` b. `k=1/sqrt2` c. `M=-1/sqrt2` d. `k=-1/sqrt2`

Options

A. `A = (1)/(3) `
B. `B = (2)/(3)`
C. `A = - (1)/(2)`
D. `B = -(1)/(6)`

Answer

To solve the integral \( I = \int \frac{\sin^2 x}{1 + \sin^2 x} \, dx \) and find the values of \( M \) and \( k \) in the expression \( I = x - k \tan^{-1}(M \tan x) \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\sin^2 x}{1 + \sin^2 x} \, dx \]
We can rewrite the numerator by adding and subtracting 1:
\[ I = \int \left( \frac{\sin^2 x + 1 - 1}{1 + \sin^2 x} \right) \, dx = \int \left( \frac{1 + \sin^2 x}{1 + \sin^2 x} - \frac{1}{1 + \sin^2 x} \right) \, dx \]
This simplifies to:
\[ I = \int 1 \, dx - \int \frac{1}{1 + \sin^2 x} \, dx \]

### Step 2: Integrate the First Part
The first integral is straightforward:
\[ \int 1 \, dx = x \]

### Step 3: Simplify the Second Integral
Now we focus on the second integral:
\[ \int \frac{1}{1 + \sin^2 x} \, dx \]
We can use the identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite it:
\[ 1 + \sin^2 x = 1 + (1 - \cos^2 x) = 2 - \cos^2 x \]
So, we have:
\[ \int \frac{1}{2 - \cos^2 x} \, dx \]

### Step 4: Use a Trigonometric Identity
We can express \( \cos^2 x \) in terms of \( \sec^2 x \):
\[ \int \frac{1}{2 - \cos^2 x} \, dx = \int \frac{1}{2 - (1 - \tan^2 x)} \, dx = \int \frac{1}{1 + \tan^2 x} \, dx \]
Using the substitution \( t = \tan x \), we have \( dt = \sec^2 x \, dx \) or \( dx = \frac{dt}{1 + t^2} \).

### Step 5: Integrate the Result
Thus, we can rewrite the integral:
\[ \int \frac{1}{1 + t^2} \, dt = \tan^{-1}(t) + C = \tan^{-1}(\tan x) \]
This leads us to:
\[ I = x - \frac{1}{\sqrt{2}} \tan^{-1}(\sqrt{2} \tan x) + C \]

### Step 6: Compare with Given Expression
Now we compare:
\[ I = x - k \tan^{-1}(M \tan x) \]
From our integration, we see that:
- \( k = \frac{1}{\sqrt{2}} \)
- \( M = \sqrt{2} \)

### Conclusion
Thus, the values are:
- \( M = \sqrt{2} \)
- \( k = \frac{1}{\sqrt{2}} \)

The correct option is:
- **b. \( k = \frac{1}{\sqrt{2}} \)**