Tính các nguyên hàm sau:
a) \(\int {x{{(3 - x)}^5}dx} \)
b) \(\int {{{({2^x} - {3^x})}^2}} dx\)
c) \(\int {x\sqrt {2 - 5x} dx} \)
d) \(\int {{{\ln (\cos x)} \over {{{\cos }^2}x}}} dx\)
e) \(\int {{x \over {{{\sin }^2}x}}} dx\)
g) \(\int {{{x + 1} \over {(x - 2)(x + 3)}}dx} \)
h) \(\int {{1 \over {1 - \sqrt x }}} dx\)
i) \(\int {\sin 3x\cos 2xdx} \)
k) \(\int {{{{{\sin }^3}x} \over {{{\cos }^2}x}}} dx\)
l) \(\int {{{\sin x\cos x} \over {\sqrt {{a^2}{{\sin }^2}x + {b^2}{{\cos }^2}x} }}} dx,({a^2} \ne {b^2})\)
HD: Đặt \(u = \sqrt {{a^2}{{\sin }^2}x + {b^2}{{\cos }^2}x} \)
Hướng dẫn làm bài
a) \({(3 - x)^6}({{3 - x} \over 7} - {1 \over 2}) + C\) .
HD: t = 3 – x
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b) \({{{4^x}} \over {\ln 4}} - 2{{{6^x}} \over {\ln 6}} + {{{9^x}} \over {\ln 9}} + C\)
c) \( - {{8 + 30x} \over {375}}{(2 - 5x)^{{3 \over 2}}} + C\).
HD: Dựa vào \(x = - {1 \over 5}(2 - 5x) + {2 \over 5}\)
d) \(\tan x{\rm{[}}\ln (\cos x) + 1] - x + C\) . HD: Đặt \(u = \ln (\cos x),dv = {{dx} \over {{{\cos }^2}x}}\)
e) \( - x\cot x + \ln |\sin x| + C\) . HD: Đặt \(u = x,dv = {{dx} \over {{{\sin }^2}x}}\)
g) \({1 \over 5}\ln [|x - 2{|^3}{(x + 3)^2}{\rm{]}} + C\)
HD: Ta có \({{x + 1} \over {(x - 2)(x + 3)}} = {3 \over {5(x - 2)}} + {2 \over {5(x + 3)}}\)
h) \( - 2(\sqrt x + \ln |1 - \sqrt x |) + C\).
HD: Đặt \(t = \sqrt x \)
i) \( - {1 \over 2}(\cos x + {1 \over 5}cos5x) + C\) .
HD: \(\sin 3x.c\cos 2x = {1 \over 2}(\sin x + \sin 5x)\)
k) \(\cos x + {1 \over {\cos x}} + C\) .
HD: Đặt u = cos x
l) \({1 \over {{a^2} - {b^2}}}\sqrt {{a^2}{{\sin }^2}x + {b^2}{{\cos }^2}x} + C\)