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Tìm
a) \(\int {\left( {{x^{{3 \over 4}}} + {x^{{1 \over 2}}} – 5} \right)} dx\)
b) \(\int {\sqrt x \left( {\sqrt x – 2x} \right)} \left( {x + 1} \right)dx\)
c) \(\int {\left( {{x^{ – 3}} – 2{x^{ – 2}} + 4x + 1} \right)} dx\)
d) \(\int {\left[ {\left( {2x + 3{x^{ – 2}}} \right)\left( {{x^2} – {1 \over x}} \right) + 3{x^{ – 3}}} \right]} dx\)
Giải
a) \(\int {\left( {{x^{{3 \over 4}}} + {x^{{1 \over 2}}} – 5} \right)} dx\) = \({4 \over 7}{x^{{7 \over 4}}} + 2{x^{{1 \over 2}}} – 5x + C\)
b) \(\int {\sqrt x \left( {\sqrt x – 2x} \right)} \left( {x + 1} \right)dx\)
\(\eqalign{
& = \int {\left( {x – 2x\sqrt x } \right)\left( {x + 1} \right)} dx \cr
& = \int {({x^2} + x – 2{x^2}\sqrt x } – 2x\sqrt x )dx \cr} \)
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\(={{{x^3}} \over 3} + {{{x^2}} \over 2} – {4 \over 7}{x^{{7 \over 2}}} – {4 \over 5}{x^{{5 \over 2}}} + C\)
c) \(\int {\left( {{x^{ – 3}} – 2{x^{ – 2}} + 4x + 1} \right)} dx\)
\( =- {1 \over {2{x^2}}} + {2 \over x} + 2{x^2} + x + C\)
d) \(\int {\left[ {\left( {2x + 3{x^{ – 2}}} \right)\left( {{x^2} – {1 \over x}} \right) + 3{x^{ – 3}}} \right]} dx\)
\(\eqalign{
& = \int {\left( {2{x^3} – 2 + 3 – {3 \over {{x^3}}} + {3 \over {{x^3}}}} \right)dx} \cr
& = \int {\left( {2{x^3} + 1} \right)dx} \cr} \)
\(={{{x^4}} \over 2} + x + C\)