Bài 25. Tính các tích phân sau :
a) \(\int\limits_0^{{\pi \over 4}} {x\cos 2xdx;} \) b) \(\int\limits_0^1 {{{\ln \left( {2 - x} \right)} \over {2 - x}}} dx;\)
c) \(\int\limits_0^{{\pi \over 2}} {{x^2}\cos xdx;} \)
\(d)\,\int\limits_0^1 {{x^2}\sqrt {{x^3} + 1} dx;} \) \(e)\,\int\limits_1^e {{x^2}\ln xdx.} \)
a) Đặt
\(\left\{ \matrix{
u = x \hfill \cr
dv = \cos 2xdx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = dx \hfill \cr
v = {1 \over 2}\sin 2x \hfill \cr} \right.\)
Do đó \(\int\limits_0^{{\pi \over 4}} {x\cos 2xdx = \left. {{1 \over 2}x\sin 2x} \right|_0^{{\pi \over 4}}} - {1 \over 2}\int\limits_0^{{\pi \over 4}} {\sin 2xdx} \)
\( = {\pi \over 8} + \left. {{1 \over 4}\cos 2x} \right|_0^{{\pi \over 4}} = {\pi \over 8} + {1 \over 4}\left( { - 1} \right) = {\pi \over 8} - {1 \over 4}.\)
b) Đặt \(u = \ln \left( {2 - x} \right) \Rightarrow du = {{ - 1} \over {2 - x}}dx\)
\(\int\limits_0^1 {{{\ln \left( {2 - x} \right)} \over {2 - x}}} dx = - \int\limits_{\ln 2}^0 {udu} = \int\limits_0^{\ln 2} {udu} = \left. {{{{u^2}} \over 2}} \right|_0^{\ln 2} = {1 \over 2}{\left( {\ln 2} \right)^2}\)
c) Đặt
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\(\left\{ \matrix{
u = {x^2} \hfill \cr
dv = \cos xdx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = 2xdx \hfill \cr
v = {\mathop{\rm s}\nolimits} {\rm{inx}} \hfill \cr} \right.\)
Do đó \(I = \int\limits_0^{{\pi \over 2}} {{x^2}\cos xdx = {x^2}} \left. {{\mathop{\rm s}\nolimits} {\rm{inx}}} \right|_0^{{\pi \over 2}} - 2\int\limits_0^{{\pi \over 2}} {x\sin xdx = {{{\pi ^2}} \over 4}} - 2{I_1}\)
Với \({I_1} = \int\limits_0^{{\pi \over 2}} {x\sin xdx} \)
Đặt
\(\left\{ \matrix{
u = x \hfill \cr
dv = \sin {\rm{x}}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = dx \hfill \cr
v = - \cos x \hfill \cr} \right.\)
Do đó \({I_1} = - x\left. {\cos x} \right|_0^{{\pi \over 2}} + \int\limits_0^{{\pi \over 2}} {\cos xdx = \left. {{\mathop{\rm s}\nolimits} {\rm{inx}}} \right|_0^{{\pi \over 2}}} = 1\)
Vậy \(I = {{{\pi ^2}} \over 4} - 2\)
d) Đặt \(u = \sqrt {{x^3} + 1} \Rightarrow {u^2} = {x^3} + 1 \Rightarrow 2udu = 3{x^2}dx \Rightarrow {x^2}dx = {2 \over 3}udu\)
\(\int\limits_0^1 {{x^2}\sqrt {{x^3} + 1} dx} = {2 \over 3}\int\limits_1^{\sqrt 2 } {{u^2}du = \left. {{{2{u^3}} \over 9}} \right|} _1^{\sqrt 2 } = {2 \over 9}\left( {2\sqrt 2 - 1} \right)\)
e) Đặt
\(\left\{ \matrix{
u = \ln x \hfill \cr
dv = {x^2}dx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = {{dx} \over x} \hfill \cr
v = {{{x^3}} \over 3} \hfill \cr} \right.\)
Do đó \(\int\limits_1^e {{x^2}\ln xdx = \left. {{{{x^3}} \over 3}\ln x} \right|} _1^e - {1 \over 3}\int\limits_1^e {{x^2}dx = {{{e^3}} \over 3} - \left. {{1 \over 9}{x^3}} \right|} _1^e = {{2{e^3} + 1} \over 9}\)