Cho hàm số f(x) = x ( 2x-1) ^ 2025 có một nguyên hàm F (x) thỏa mãn

18/22

Cho hàm số \(f\left( x \right) = x{\left( {2x - 1} \right)^{2025}}\) có một nguyên hàm \(F\left( x \right)\) thỏa mãn \[F\left( {\frac{1}{2}} \right) = 1\]. Tính \[F\left( 0 \right)\] (làm tròn đến hàng đơn vị).

Giải thích

+) Ta có \(f\left( x \right) = x{\left( {2x - 1} \right)^{2025}}\)\( = \frac{1}{2}\left( {2x - 1 + 1} \right){\left( {2x - 1} \right)^{2025}}\)\( = \frac{1}{2}{\left( {2x - 1} \right)^{2026}} + \frac{1}{2}{\left( {2x - 1} \right)^{2025}}\)..

+) \(F\left( x \right) = \int {f\left( x \right){\rm{d}}x} \)\( = \int {\left( {\frac{1}{2}{{\left( {2x - 1} \right)}^{2026}} + \frac{1}{2}{{\left( {2x - 1} \right)}^{2025}}} \right){\rm{d}}x} \)\( = \frac{1}{4}\frac{{{{\left( {2x - 1} \right)}^{2027}}}}{{2027}} + \frac{1}{4}\frac{{{{\left( {2x - 1} \right)}^{2026}}}}{{2026}} + C\)\( = \frac{{{{\left( {2x - 1} \right)}^{2027}}}}{{8108}} + \frac{{{{\left( {2x - 1} \right)}^{2026}}}}{{8104}} + C\).

Lại có: \[F\left( {\frac{1}{2}} \right) = 1\] \[ \Rightarrow C = 1\]\( \Rightarrow F\left( x \right) = \frac{{{{\left( {2x - 1} \right)}^{2027}}}}{{8108}} + \frac{{{{\left( {2x - 1} \right)}^{2026}}}}{{8104}} + 1\)\( \Rightarrow F\left( 0 \right) = \frac{{{{\left( { - 1} \right)}^{2027}}}}{{8108}} + \frac{{{{\left( { - 1} \right)}^{2026}}}}{{8104}} + 1 \approx 1\).

Vậy \[F\left( 0 \right) \approx 1\].