อ้างอิง:
ข้อความเดิมเขียนโดยคุณ Mastermander
โอเคครับ เอาง่ายๆลงมาหน่อย
\[
\int {\frac{{x^2 - x^3 }}{{1 - x^3 }}dx}
\]
|
\[
\int {\frac{{x^2 - x^3 }}{{1 - x^3 }}dx} = \int {\frac{{x^2 }}{{x^2 + x + 1}}dx}
\]
\[
\int {\frac{{x^2 - x^3 }}{{1 - x^3 }}dx} = \int {1 - \frac{{x + 1}}{{x^2 + x + 1}}dx} = x - \int {\frac{{x + 1}}{{x^2 + x + 1}}dx}
\]
\[
= x - \frac{1}{2}\left( {\int {\frac{{2x + 1}}{{x^2 + x + 1}} + \frac{1}{{x^2 + x + 1}}dx} } \right)
\]
\[
= x - \frac{1}{2}\left( {\int {\frac{{d\left( {x^2 + x + 1} \right)}}{{x^2 + x + 1}} + \int {\frac{1}{{\left( {x + \frac{1}{2}} \right)^2 + \frac{3}{4}}}d\left( {x + \frac{1}{2}} \right)} } } \right)
\]
\[
\int {\frac{{x^2 - x^3 }}{{1 - x^3 }}dx} = x - \frac{1}{2}\left( {\ln \left( {x^2 + x + 1} \right) + \frac{1}{{\sqrt 3 }}\arctan \left( {\frac{{2x + 1}}{{\sqrt 3 }}} \right)} \right)
\]