Cho ∫ f ( x ) dx = x^2 + x + C1 , ∫ g ( x ) dx = x 4 + x^3 + C2 . a) f ( x ) = 2x + 1 .

12/20

Cho \(\int {f\left( x \right)dx} = {x^2} + x + {C_1},\int {g\left( x \right)dx} = {x^4} + {x^3} + {C_2}\).

a) \(f\left( x \right) = 2x + 1\).

b) \(g\left( 0 \right) = 1\).

c) \(\int\limits_0^1 {g\left( x \right)dx} = 3\).

d) \(\int\limits_0^1 {f\left( x \right)g\left( x \right)dx} = \frac{{51}}{{10}}\).

Nhập câu trả lời ở đây

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Giải thích

a)\(f\left( x \right) = {\left( {{x^2} + x + C} \right)^\prime } = 2x + 1\).

b) \(g\left( x \right) = {\left( {{x^4} + {x^3} + C} \right)^\prime } = 4{x^3} + 3{x^2}\). Suy ra \(g\left( 0 \right) = 0\).

c) \(\int\limits_0^1 {g\left( x \right)dx} = \int\limits_0^1 {\left( {4{x^3} + 3{x^2}} \right)dx} = \left. {\left( {{x^4} + {x^3}} \right)} \right|_0^1 = 2\).

d) \(\int\limits_0^1 {f\left( x \right)g\left( x \right)dx} = \int\limits_0^1 {\left( {2x + 1} \right)\left( {4{x^3} + 3{x^2}} \right)dx} \)\( = \int\limits_0^1 {\left( {8{x^4} + 10{x^3} + 3{x^2}} \right)dx} \)\( = \left. {\left( {\frac{8}{5}{x^5} + \frac{{10}}{4}{x^4} + {x^3}} \right)} \right|_0^1 = \frac{{51}}{{10}}\).

Đáp án: a) Đúng; b) Sai; c) Sai; d) Đúng.