Câu 3.4 trang 141 sách bài tập Giải tích 12 Nâng cao

Tìm

Tìm

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

b) (int {sqrt x left( {sqrt x  - 2x} ight)} left( {x + 1} ight)dx)

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

d) (int {left[ {left( {2x + 3{x^{ - 2}}} ight)left( {{x^2} - {1 over x}} ight) + 3{x^{ - 3}}} ight]} dx)

Giải

a) (int {left( {{x^{{3 over 4}}} + {x^{{1 over 2}}} - 5} ight)} dx)  = ({4 over 7}{x^{{7 over 4}}} + 2{x^{{1 over 2}}} - 5x + C)

b) (int {sqrt x left( {sqrt x  - 2x} ight)} left( {x + 1} ight)dx)

(eqalign{
& = int {left( {x - 2xsqrt x } ight)left( {x + 1} ight)} dx cr
& = int {({x^2} + x - 2{x^2}sqrt x } - 2xsqrt x )dx cr} )

  (={{{x^3}} over 3} + {{{x^2}} over 2} - {4 over 7}{x^{{7 over 2}}} - {4 over 5}{x^{{5 over 2}}} + C)

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

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

d) (int {left[ {left( {2x + 3{x^{ - 2}}} ight)left( {{x^2} - {1 over x}} ight) + 3{x^{ - 3}}} ight]} dx)

(eqalign{
& = int {left( {2{x^3} - 2 + 3 - {3 over {{x^3}}} + {3 over {{x^3}}}} ight)dx} cr
& = int {left( {2{x^3} + 1} ight)dx} cr} )

  (={{{x^4}} over 2} + x + C)

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