Question

Evaluate the integral \( \displaystyle \int \frac{e^x}{\sqrt{x}}(1+2x)\,dx \)

• A. \( \dfrac{1}{\sqrt{x}} e^x + c \)
• B. \( 2\sqrt{x}\, e^x + c \)
• C. \( \dfrac{\sqrt{x}}{2} e^x + c \)
• D. \( \sqrt{x}\, e^x + c \)

Hint:

When an integrand looks complicated, try differentiating the options. This often leads directly to the correct answer.

Answer 1

The Correct Option is B

Step 1: Observe the structure of the integrand.
We notice that \[ \frac{e^x}{\sqrt{x}}(1+2x) \] resembles the derivative of a product involving \( \sqrt{x} \) and \( e^x \).
Step 2: Try differentiation of \( 2\sqrt{x}\, e^x \).
\[ \frac{d}{dx}(2\sqrt{x}\, e^x) \] Step 3: Apply the product rule.
\[ = 2\left( \frac{1}{2\sqrt{x}} e^x + \sqrt{x} e^x \right) \] \[ = \frac{e^x}{\sqrt{x}} + 2\sqrt{x} e^x \] Step 4: Factor out \( \dfrac{e^x}{\sqrt{x}} \).
\[ = \frac{e^x}{\sqrt{x}}(1 + 2x) \] Step 5: Conclusion.
Thus, \[ \int \frac{e^x}{\sqrt{x}}(1+2x)\,dx = 2\sqrt{x}\, e^x + c \]