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Bài 18. Dùng phương pháp tích phân từng phần để tính các tích phân sau:
a) \(\int\limits_1^2 {{x^5}} \ln xdx;\) b) \(\int\limits_0^1 {\left( {x + 1} \right)} {e^x}dx;\)
c) \(\int\limits_0^\pi {{e^x}} \cos xdx;\) d) \(\int\limits_0^{{\pi \over 2}} {x\cos xdx.} \)
a) Đặt
\(\left\{ \matrix{
u = \ln x \hfill \cr
dv = {x^5}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = {{dx} \over x} \hfill \cr
v = {{{x^6}} \over 6} \hfill \cr} \right.\)
\(\int\limits_1^2 {{x^5}} \ln xdx = \left. {{{{x^6}} \over 6}\ln x} \right|_1^2 – {1 \over 6}\int\limits_1^2 {{x^5}} dx = \left. {\left( {{{{x^6}} \over 6}\ln x – {{{x^6}} \over {36}}} \right)} \right|_1^2 = {{32} \over 3}\ln 2 – {7 \over 4}\)
b) Đặt
\(\left\{ \matrix{
u = x + 1 \hfill \cr
dv = {e^x}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = dx \hfill \cr
v = {e^x} \hfill \cr} \right.\)
\(\int\limits_0^1 {\left( {x + 1} \right)} {e^x}dx = \left. {\left( {x + 1} \right){e^x}} \right|_0^1 – \int\limits_0^1 {{e^x}dx = e} \)
c) Đặt \(I = \int\limits_0^\pi {{e^x}\cos xdx} \)
Đặt
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\(\left\{ \matrix{
u = {e^x} \hfill \cr
dv = \cos xdx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = {e^x}dx \hfill \cr
v = {\mathop{\rm s}\nolimits} {\rm{inx}} \hfill \cr} \right.\)
Suy ra \(I = \left. {{e^x}{\mathop{\rm s}\nolimits} {\rm{inx}}} \right|_0^\pi – \int\limits_0^\pi {{e^x}\sin {\rm{x}}dx} = – \int\limits_0^\pi {{e^x}\sin {\rm{x}}dx} \)
Đặt
\(\left\{ \matrix{
u = {e^x} \hfill \cr
dv = \sin {\rm{x}}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = {e^x}dx \hfill \cr
v = – \cos x \hfill \cr} \right.\)
Do đó \(I = – \left[ {\left. {\left( { – {e^x}\cos x} \right)} \right|_0^\pi + \int\limits_0^\pi {{e^x}\cos xdx} } \right] = {e^\pi }\cos \pi – {e^0}.\cos 0 – I\)
\( \Rightarrow 2I = – {e^\pi } – 1 \Rightarrow I = – {1 \over 2}\left( {{e^\pi } + 1} \right)\)
b) Đặt
\(\left\{ \matrix{
u = x \hfill \cr
dv = \cos xdx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = dx \hfill \cr
v = {\mathop{\rm s}\nolimits} {\rm{inx}} \hfill \cr} \right.\)
Do đó \(\int\limits_0^{{\pi \over 2}} {x\cos xdx = \left. {x\sin x} \right|_0^{{\pi \over 2}}} – \int\limits_0^{{\pi \over 2}} {\sin {\rm{x}}dx = \left. {\left( {x\sin x + \cos x} \right)} \right|_0^{{\pi \over 2}}} = {\pi \over 2} – 1\)