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 \( f(x) \)
\[f'(x) = \frac{\sqrt{1+x^2} \cdot 1 - x \cdot \frac{1}{2\sqrt{1+x^2}} \cdot 2x}{(1+x^2)}.\]Simplify the numerator:\[f'(x) = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{1+x^2} = \frac{\frac{1+x^2-x^2}{\sqrt{1+x^2}}}{1+x^2} = \frac{1}{(1+x^2)^{3/2}}.\]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 the rule:\[\int e^x \left( f(x) + f'(x) \right) dx = e^x f(x) + C,\]we substitute \( f(x) = \frac{x}{\sqrt{1+x^2}} \), yielding:\[I = e^x \frac{x}{\sqrt{1+x^2}} + C.\]
Final Answer:\[\boxed{I = e^x \frac{x}{\sqrt{1+x^2}} + C.}\]
Explanation:
1. Rewrite the integral in the form \( \int e^x (f(x) + f'(x)) dx \).2. Identify \( f(x) = \frac{x}{\sqrt{1+x^2}} \).3. Verify that \( f'(x) = \frac{1}{(1+x^2)^{3/2}} \).4. Apply the integration formula \( \int e^x (f(x) + f'(x)) dx = e^x f(x) + C \).5. Substitute \( f(x) \) to obtain the final result.