Statement I `int 2^(tan^(-1)x)d(cot^(-1)x)=(2^(tan^(-1)x))/(ln 2)+C`
Statement II `(d)/(dx) (a^(x)+C)=a^(x) ln a`

Options

A. Statement I is true, Statement II is also true , Statement II is the correct explanation of Statement I.
B. Statement I is true, Statement II is also true, Statement II is not the correct explanation of Statement I.
C. Statement I is true, Statement II is false.
D. Statement I is false, Statement II is true .

Answer

Let's solve the integral step by step.

### Given Problem:
We need to verify the statement:
\[ \int 2^{\tan^{-1} x} \, d(\cot^{-1} x) = \frac{2^{\tan^{-1} x}}{\ln 2} + C \]

### Step 1: Change of Variable
Let \( T = \cot^{-1} x \). Then, we have:
\[ \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1+x^2} \implies d(\cot^{-1} x) = -\frac{1}{1+x^2} \, dx \]
Thus, the integral becomes:
\[ \int 2^{\tan^{-1} x} \, d(\cot^{-1} x) = -\int 2^{\tan^{-1} x} \frac{1}{1+x^2} \, dx \]

### Step 2: Express \(\tan^{-1} x\) in terms of \(T\)
Using the identity:
\[ \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2} \]
We can express \(\tan^{-1} x\) as:
\[ \tan^{-1} x = \frac{\pi}{2} - T \]

### Step 3: Substitute in the Integral
Substituting this into the integral, we have:
\[ -\int 2^{\frac{\pi}{2} - T} \frac{1}{1+x^2} \, dx \]
This can be rewritten as:
\[ -\int \frac{2^{\frac{\pi}{2}}}{2^T} \frac{1}{1+x^2} \, dx \]

### Step 4: Solve the Integral
Now, we can factor out the constant \(2^{\frac{\pi}{2}}\):
\[ -2^{\frac{\pi}{2}} \int \frac{1}{2^T(1+x^2)} \, dx \]
The integral \(\int \frac{1}{1+x^2} \, dx\) gives \(\tan^{-1} x\).

### Step 5: Final Expression
Thus, we have:
\[ -2^{\frac{\pi}{2}} \left( \tan^{-1} x \cdot \frac{1}{2^T} \right) + C \]
Substituting back \(T = \cot^{-1} x\):
\[ = -\frac{2^{\frac{\pi}{2}} \tan^{-1} x}{2^{\cot^{-1} x}} + C \]
This does not match the original statement.

### Conclusion
The statement:
\[ \int 2^{\tan^{-1} x} \, d(\cot^{-1} x) = \frac{2^{\tan^{-1} x}}{\ln 2} + C \]
is incorrect.

### Verification of Statement II
The second statement is:
\[ \frac{d}{dx} (a^x + C) = a^x \ln a \]
This is true as the derivative of a constant \(C\) is zero.

### Final Result
- Statement I is **False**.
- Statement II is **True**.