Giải bài 24 trang 162 SGK giải tích nâng cao 12
Tính các tích phân sau:
a) \(\int\limits_{1}^{2}{{{x}^{2}}{{e}^{{{x}^{3}}}}dx};\) b) \(\int\limits_{1}^{3}{\dfrac{1}{x}{{\left( \ln x \right)}^{2}}dx}\);
c) \(\int\limits_{0}^{\sqrt{3}}{x\sqrt{1+{{x}^{2}}}dx}\); d) \(\int\limits_{0}^{1}{{{x}^{2}}{{e}^{3{{x}^{3}}}}dx}\); e) \(\int\limits_{0}^{\frac{\pi }{2}}{\dfrac{\cos x}{1+\sin x}dx} \).
a) Đặt \({{x}^{3}}=t\Rightarrow 3{{x}^{2}}dx=dt\)
| x | 1 | 2 |
| t | 1 | 8 |
\(\Rightarrow \int\limits_{1}^{2}{{{x}^{2}}{{e}^{{{x}^{3}}}}dx}=\dfrac{1}{3}\int\limits_{1}^{8}{{{e}^{t}}dt}=\dfrac{1}{3}{{e}^{t}}\left| _{\begin{smallmatrix} \\ \\\\ 1 \end{smallmatrix}}^{\begin{smallmatrix} 8 \\ \\\\ \end{smallmatrix}} \right.=\dfrac{1}{3}\left( {{e}^{8}}-e \right) \)
b) Đặt \(\ln x=t\Rightarrow \dfrac{1}{x}dx=dt\)
| x | 1 | 3 |
| t | 0 | \(\ln 3\) |
\(\Rightarrow \int\limits_{1}^{3}{\dfrac{1}{x}{{\left( \ln x \right)}^{2}}dx}=\int\limits_{0}^{\ln 3}{{{t}^{2}}dt}=\dfrac{1}{3}{{t}^{3}}\left| _{\begin{smallmatrix} \\ \\\\ 0 \end{smallmatrix}}^{\begin{smallmatrix} \ln 3 \\\\\\ \end{smallmatrix}}=\dfrac{{{\left( \ln 3 \right)}^{3}}}{3} \right. \)
c) Đặt \(\sqrt{1+{{x}^{2}}}=t\Rightarrow 1+{{x}^{2}}={{t}^{2}}\Rightarrow xdx=tdt \)
| x | 0 | \(\sqrt 3\) |
| t | 1 | 2 |
\(\Rightarrow \int\limits_{0}^{\sqrt{3}}{x\sqrt{1+{{x}^{2}}}dx}=\int\limits_{1}^{2}{t^2dt}=\dfrac{1}{3}{{t}^{3}}\left| _{\begin{smallmatrix} \\ \\\\ 1 \end{smallmatrix}}^{\begin{smallmatrix} 2 \\ \\\\ \end{smallmatrix}} \right.=\dfrac{1}{3}\left( 8-1 \right)=\dfrac{7}{3} \)
d) Đặt \(3{{x}^{3}}=t\Rightarrow 9{{x}^{2}}dx=dt\)
| x | 0 | 1 |
| t | 0 | 3 |
\(\Rightarrow \int\limits_{0}^{1}{{{x}^{2}}{{e}^{3{{x}^{3}}}}dx}=\dfrac{1}{9}\int\limits_{0}^{3}{{{e}^{t}}dt}=\dfrac{1}{9}{{e}^{t}}\left| _{\begin{smallmatrix} \\ \\\\ 0 \end{smallmatrix}}^{\begin{smallmatrix} 3 \\ \\\\ \end{smallmatrix}} \right.=\dfrac{1}{9}\left( {{e}^{3}}-1 \right) \)
e) Đặt \(1+\sin x=t\Rightarrow \cos xdx=dt \)
| x | 1 | \(\dfrac{\pi}{2}\) |
| t | 1 | 2 |
\(\Rightarrow \int\limits_{0}^{\frac{\pi }{2}}{\dfrac{\cos x}{1+\sin x}dx}=\int\limits_{1}^{2}{\dfrac{dt}{t}}=\ln \left| t \right|\left| _{\begin{smallmatrix} \\ \\ 1 \end{smallmatrix}}^{\begin{smallmatrix} 2 \\ \\ \end{smallmatrix}} \right.=\ln 2 \)