Bài 42
a) \(y = {1 \over {{x^2}}}\cos \left( {{1 \over x} – 1} \right)\); b) \(y = {x^3}{\left( {1 + {x^4}} \right)^3}\);
c) \(y = {{x{e^{2x}}} \over 3}\); d) \(y = {x^2}{e^x}\).
a) Đặt \(u = {1 \over x} – 1 \Rightarrow du = – {1 \over {{x^2}}}dx \Rightarrow {{dx} \over {{x^2}}} = – du\)
Do đó \(\int {{1 \over {{x^2}}}} \cos \left( {{1 \over x} – 1} \right)dx = – \int {\cos udu = – \sin u + C = – \sin \left( {{1 \over x} – 1} \right)} + C\)
b) Đặt \(u = 1 + {x^4} \Rightarrow du = 4{x^3}dx \Rightarrow {x^3}dx = {{du} \over 4}\)
\(\int {{x^3}{{\left( {1 + {x^4}} \right)}^3}dx = {1 \over 4}\int {{u^3}du = {{{u^4}} \over {16}} + C = {1 \over {16}}} } {\left( {1 + {x^4}} \right)^4} + C\)
c) Đặt
\(\left\{ \matrix{
u = {x \over 3} \hfill \cr
dv = {e^{2x}}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = {1 \over 3}dx \hfill \cr
v = {1 \over 2}{e^{2x}} \hfill \cr} \right.\)
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Suy ra: \(\int {{{x{e^{2x}}} \over 3}dx = {1 \over 6}x{e^{2x}} – {1 \over 6}\int {{e^{2x}}dx = {1 \over 6}x{e^{2x}} – {1 \over {12}}{e^{2x}} + C} } \)
d) Đặt
\(\left\{ \matrix{
u = {x^2} \hfill \cr
dv = {e^x}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = 2xdx \hfill \cr
v = {e^x} \hfill \cr} \right.\)
Suy ra \(\int {{x^2}{e^x}dx = {x^2}{e^x} – 2\int {x{e^x}dx} } \) (1)
Đặt
\(\left\{ \matrix{
u = x \hfill \cr
dv = {e^x}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = dx \hfill \cr
v = {e^x} \hfill \cr} \right.\)
Do đó: \(\int {x{e^x}dx = x{e^x} – \int {{e^x}dx = x{e^x} – {e^x} + C} } \)
Từ (1) suy ra \(\int {{x^2}{e^x}dx = {x^2}{e^x} – 2x{e^x} + 2{e^x} + C = {e^x}\left( {{x^2} – 2x + 2} \right) + C} \)