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: Rewrite the given integral
The integral is: \[ I = \int e^x \left( \frac{x}{\sqrt{1+x^2}} + \frac{1}{(1+x^2)^{3/2}} \right) dx. \]
Let: \[ f(x) = \frac{x}{\sqrt{1+x^2}}. \]
Step 2: Differentiate \( f(x) \)
\[ f'(x) = \frac{\sqrt{1+x^2} - \frac{x \cdot x}{\sqrt{1+x^2}}}{1+x^2}. \] Simplify the numerator: \[ f'(x) = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{(1+x^2)} = \frac{1}{(1+x^2)^{3/2}}. \] Thus, the integral can be rewritten as: \[ I = \int e^x \left( f(x) + f'(x) \right) dx. \]
Step 3: Apply the standard integral result
Using: \[ \int e^x \left( f(x) + f'(x) \right) dx = e^x f(x) + C, \] we substitute \( f(x) = \frac{x}{\sqrt{1+x^2}} \), giving: \[ I = e^x \frac{x}{\sqrt{1+x^2}} + C. \]
Final Answer: \[ \boxed{I = e^x \frac{x}{\sqrt{1+x^2}} + C.} \]
Explanation:
1. Splitting the Integral: The given integral is broken into two terms. 2. Choosing \( f(x) \): We set \( f(x) = \frac{x}{\sqrt{1+x^2}} \), ensuring that its derivative matches the second term. 3. Applying the Formula: Using the integral result for \( e^x (f(x) + f'(x)) \), we directly obtain the solution. 4. Substituting \( f(x) \): Finally, substituting \( f(x) \) yields the required answer.