Tìm nguyên hàm I = x^2 -2025 / x ( x^2 + 2025 ) dx

9/22

Tìm nguyên hàm \(I = \int {\frac{{{x^2} - 2025}}{{x\left( {{x^2} + 2025} \right)}}} {\rm{d}}x\).

\(I = \ln \left| {x - \frac{{2025}}{x}} \right| + C\).

\(I = \ln \left| {1 - \frac{{2025}}{x}} \right| + C\).

\(I = \ln \left| {x + \frac{{2025}}{x}} \right| + C\).

\(I = \ln \left| {1 + \frac{{2025}}{x}} \right| + C\).

Giải thích

Ta có: \(I = \int {\frac{{{x^2} - 2025}}{{x\left( {{x^2} + 2025} \right)}}} {\rm{d}}x\)\( = \int {\frac{{1 - \frac{{2025}}{{{x^2}}}}}{{x + \frac{{2025}}{x}}}} {\rm{d}}x\)\( = \int {\frac{1}{{\left( {x + \frac{{2025}}{x}} \right)}}{\rm{d}}\left( {x + \frac{{2025}}{x}} \right)} \)

Vậy \(I = \ln \left| {x + \frac{{2025}}{x}} \right| + C\).