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Tính các tích phân sau:
a) \(\int\limits_0^1 {({y^3} + 3{y^2} – 2)dy} \)
b)\(\int\limits_1^4 {(t + {1 \over {\sqrt t }}} – {1 \over {{t^2}}})dt\)
c) \(\int\limits_0^{{\pi \over 2}} {(2\cos x – \sin 2x)dx} \)
d) \(\int\limits_0^1 {{{({3^s} – {2^s})}^2}ds} \)
e) \(\int\limits_0^{{\pi \over 3}} {\cos 3xdx} + \int\limits_{{\pi \over 3}}^{{{3\pi } \over 2}} {\cos 3xdx} + \int\limits_{{{3\pi } \over 2}}^{{{5\pi } \over 2}} {\cos 3xdx} \)
g)\(\int\limits_0^3 {|{x^2} – x – 2|dx} \)
h) \(\int\limits_\pi ^{{{5\pi } \over 4}} {{{\sin x – \cos x} \over {\sqrt {1 + \sin 2x} }}} dx\)
i) \(\int\limits_0^4 {{{4x – 1} \over {\sqrt {2x + 1} + 2}}} dx\)
Hướng dẫn làm bài
a) \( – {3 \over 4}\)
b) \({{35} \over 4}\)
c) 1
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d) \({4 \over {\ln 3}} – {{10} \over {\ln 6}} + {3 \over {2\ln 2}}\)
e) \( – {1 \over 3}\)
g) \({{31} \over 6}\) .
HD: \(\int\limits_0^3 {|{x^2} – x – 2|dx }\)
\({= \int\limits_0^2 { – ({x^2} – x – 2)dx + \int\limits_2^3 {({x^2} – x – 2)dx} } } \)
h) \({1 \over 2}\ln 2\) .
HD: \(\int\limits_\pi ^{{{5\pi } \over 4}} {{{\sin x – \cos x} \over {\sqrt {1 + \sin 2x} }}} dx\)
\(= \int\limits_\pi ^{{{5\pi } \over 4}} {{{\sin x – \cos x} \over {|\sin x + \cos x|}}} dx = \int\limits_\pi ^{{{5\pi } \over 4}} {{{d(\sin x + \cos x)} \over {\sin x + \cos x}}} \)
i) \({{34} \over 3} + 10\ln {3 \over 5}\) .
HD: Đặt \(t = \sqrt {2x + 1} \)
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