Question

Find: \( \int e^x \left[ \frac{1}{(1 + x^2)^{3/2}} + \frac{x}{\sqrt{1 + x^2}} \right] dx \).

Hint:

For integrals involving a combination of functions \( f(x) \) and \( f'(x) \), use the substitution \( u = f(x) \) to simplify.

Answer 1

Step 1: The given integral can be written as: \[ I = \int e^x \left( \frac{x}{\sqrt{1+x^2}} + \frac{1}{(1+x^2)^{\frac{3}{2}}} \right) dx \] Let: \[ f(x) = \frac{x}{\sqrt{1+x^2}} \] Step 2: Now, calculate the derivative of \( f(x) \): \[ f'(x) = \frac{\sqrt{1+x^2} - \frac{x \cdot x}{\sqrt{1+x^2}}}{1+x^2} = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{1+x^2} \] Simplify the numerator: \[ f'(x) = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{1+x^2} = \frac{1}{(1+x^2)^{\frac{3}{2}}} \] Thus, the integral becomes: \[ I = \int e^x \left( f(x) + f'(x) \right) dx \] Step 3: Using the standard result: \[ \int e^x \left( f(x) + f'(x) \right) dx = e^x f(x) + C \] Substitute \( f(x) = \frac{x}{\sqrt{1+x^2}} \): \[ I = e^x \frac{x}{\sqrt{1+x^2}} + C \] Final Answer: \[ \boxed{I = e^x \frac{x}{\sqrt{1+x^2}} + C} \] Explanation: 1. Splitting the Integral: The given integral is split into terms containing \( \frac{x}{\sqrt{1+x^2}} \) and \( \frac{1}{(1+x^2)^{\frac{3}{2}}} \). 
2. Defining \( f(x) \): The function \( f(x) \) is chosen as \( \frac{x}{\sqrt{1+x^2}} \) because its derivative results in the second term, \( \frac{1}{(1+x^2)^{\frac{3}{2}}} \). 
3. Applying the Formula: The integral formula for \( \int e^x (f(x) + f'(x)) dx = e^x f(x) + C \) is directly applied. 
4. Substitution: Finally, substituting \( f(x) \) into the formula gives the result.