The value of integral `inte^x(1/(sqrt(1+x^2))+1/(sqrt((1+x^2)^5)))dxi se q u a lto` `e^x(1/(sqrt(1+x^2))+1/(sqrt((1+x^2)^3)))+c` `e^x(1/(sqrt(1+x^2))-1/(sqrt((1+x^2)^3)))+c` `e^x(1/(sqrt(1+x^2))+1/(sqrt((1+x^2)^5)))+c` none of these

Options

A. ` e^(x)((1)/(sqrt(1+x^(2)))+(x)/(sqrt((1+x^(2))^(3))))+c`
B. ` e^(x)((1)/(sqrt(1+x^(2)))-(x)/(sqrt((1+x^(2))^(3))))+c`
C. ` e^(x)((1)/(sqrt(1+x^(2)))+(x)/(sqrt((1+x^(2))^(5))))+c`
D. none of these

Explanation

To solve the integral

\[ \int e^x \left( \frac{1}{\sqrt{1+x^2}} + \frac{1}{\sqrt{(1+x^2)^5}} \right) dx, \]

we can use the technique of integration by recognizing a pattern that resembles the derivative of a function multiplied by \( e^x \).

### Step 1: Identify the function and its derivative
Let’s denote:

\[ f(x) = \frac{1}{\sqrt{1+x^2}}. \]

Now, we need to find the derivative \( f'(x) \):

\[ f'(x) = \frac{d}{dx} \left( (1+x^2)^{-1/2} \right) = -\frac{1}{2} (1+x^2)^{-3/2} \cdot (2x) = -\frac{x}{(1+x^2)^{3/2}}. \]

### Step 2: Combine terms
Now, we can express our integral in terms of \( f(x) \) and its derivative. The second term in the integral can be rewritten using the derivative we found:

\[ \frac{1}{\sqrt{(1+x^2)^5}} = (1+x^2)^{-5/2} = (1+x^2)^{-3/2} \cdot (1+x^2)^{-1} = f'(x) \cdot (1+x^2)^{-1}. \]

### Step 3: Rewrite the integral
Thus, we can rewrite the integral as:

\[ \int e^x \left( f(x) + f'(x) \cdot (1+x^2)^{-1} \right) dx. \]

### Step 4: Apply integration by parts
Now, we can apply the integration formula for \( e^x f(x) \):

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

### Step 5: Solve the integral
Now we can integrate:

\[ \int e^x \left( \frac{1}{\sqrt{1+x^2}} + \frac{1}{\sqrt{(1+x^2)^5}} \right) dx = e^x \left( \frac{1}{\sqrt{1+x^2}} + \frac{1}{\sqrt{(1+x^2)^3}} \right) + C. \]

### Final Answer
Thus, the final answer is:

\[ e^x \left( \frac{1}{\sqrt{1+x^2}} + \frac{1}{\sqrt{(1+x^2)^3}} \right) + C. \]