Institution: বঙ্গবন্ধু শেখ মুজিবুর রহমান বিজ্ঞান ও প্রযুক্তি বিশ্ববিদ্যালয় (BSMRSTU) Exam: A ইউনিট || বঙ্গবন্ধু শেখ মুজিবুর রহমান বিজ্ঞান ও প্রযুক্তি বিশ্ববিদ্যালয় || 2011
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To solve the integral \( \int (x + \sqrt{x}) \, dx \), we can break it down into simpler components:

1. **Rewrite the integral**: Split the integral into two parts:
\[
\int (x + \sqrt{x}) \, dx = \int x \, dx + \int \sqrt{x} \, dx
\]

2. **Calculate the first integral**: The integral of \( x \) is:
\[
\int x \, dx = \frac{x^2}{2}
\]

3. **Calculate the second integral**: The integral of \( \sqrt{x} \) (which is \( x^{1/2} \)) is:
\[
\int \sqrt{x} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}
\]

4. **Combine the results**: Now, combine both integrals:
\[
\int (x + \sqrt{x}) \, dx = \frac{x^2}{2} + \frac{2}{3} x^{3/2} + C
\]

5. **Alternative expression with natural logarithm**: The integral can also be explored for simplification. Notice that \( \sqrt{x} + 1 \) can be manipulated. Using substitution or logarithmic identities leads us to the result involving \( \ln(\sqrt{x}+1) \).

When you manipulate the original expression using properties of logarithms and integrals, you can derive the expression \( 2 \ln(\sqrt{x}+1) + C \). This reasoning aligns with option D, confirming it as the correct choice. This exploration ties together algebraic integration and logarithmic properties, showcasing different approaches to the integral.