Đáp án:

• \( \int \frac{1}{x(x^3 + 1)} \, dx = \int \frac{x^2}{x^3(x^3 + 1)} \, dx \)

• Đặt \( t = x^3 \implies dt = 3x^2 \, dx \)

• \( \int \frac{x^2}{x^3(x^3 + 1)} \, dx = \frac{1}{3} \int \frac{1}{t(t + 1)} \, dt \)

\( = \frac{1}{3} \int \left( \frac{1}{t} - \frac{1}{t + 1} \right) \, dt = \frac{1}{3} \ln \left| \frac{t}{t + 1} \right| + C \)

Cách 2: Thêm bớt

• \( \int \frac{1}{x(x^3 + 1)} \, dx = \int \frac{1 + x^3 - x^3}{x(x^3 + 1)} \, dx \)

\( = \int \frac{1}{x} \, dx - \int \frac{x^2}{x^3 + 1} \, dx. \)