Bài tập: Tính \( I = \int \frac{x}{3x - \sqrt{9x^2 - 1}} \, dx \)
A. \( I = \frac{1}{27} \left( 9x^2 + 1 \right)^{\frac{3}{2}} + x^3 + C \)
B. \( I = \frac{1}{27} \left( 9x^2 - 1 \right)^{\frac{3}{2}} + \frac{x^3}{3} + C \)
C. \( I = \frac{1}{27} \left( 9x^2 - 1 \right)^{\frac{3}{2}} + x^3 + C \)
D. \( I = \frac{1}{54} \left( 9x^2 - 1 \right)^{\frac{3}{2}} + x^3 + C \)
Đáp án:
• \( I = \int x \left( 3x + \sqrt{9x^2 - 1} \right) \, dx = \int \left( 3x^2 + x \sqrt{9x^2 - 1} \right) \, dx \)
\( = x^3 + \frac{1}{18} \cdot \frac{\left( 9x^2 - 1 \right)}{\frac{3}{2}}^{\frac{3}{2}} + C = x^3 + \frac{1}{27} \left( 9x^2 - 1 \right)^{\frac{3}{2}} + C \)
\(\Rightarrow\) Vậy chọn đáp án \( \boxed{\text{C}} \)