Câu 3.3 trang 141 sách bài tập Giải tích 12 Nâng cao

Tìm

Tìm

a) (int {{{{x^2} - 3x} over x}} dx)                  b) (int {{{4{x^3} + 5x - 1} over {{x^2}}}} dx)

c) (int {{{{{left( {x + 2} ight)}^2}} over {{x^4}}}} dx)                  d) (int {{{{{left( {{x^2} + 1} ight)}^2}} over {{x^2}}}} dx)

Giải

a) ( int {{{{x^2} - 3x} over x}} dx= int {(x - 3)} dx={{{x^2}} over 2} - 3x + C)                    

b) (int {{{4{x^3} + 5x - 1} over {{x^2}}}} dx= int {left( {4x + {5 over x} - {1 over {{x^2}}}} ight)} dx)

(=2{x^2} + 2x + {1 over x} + C)

c) ( int {{{{{left( {x + 2} ight)}^2}} over {{x^4}}}} dx = int {{{{x^2} + 4x + 4} over {{x^4}}}}  = int {left( {{1 over {{x^2}}} + {4 over {{x^3}}} + {4 over {{x^4}}}} ight)} dx)

(=- {1 over x} - {2 over {{x^2}}} - {4 over {3{x^3}}} + C)    

d) (int {{{{{left( {{x^2} + 1} ight)}^2}} over {{x^2}}}} dx = int {{{{x^4} + 2{x^2} + 1} over {{x^4}}}} )

(= int {left( {1 + {2 over {{x^2}}} + {1 over {{x^4}}}} ight)} dx={{{x^3}} over 3} + {{{x^2}} over 2} - {1 over x} + C)

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