Bài tập 24 trang 162 SGK Toán 12 NC

Bài tập 24 trang 162 SGK Toán 12 NC

a) Đặt \(u = {x^3} \Rightarrow du = 3{x^2}dx \)

\(\Rightarrow {x^2}dx = \frac{{du}}{3}\)

\(\begin{array}{l}
\int\limits_1^2 {{x^2}{e^{{x^3}}}dx}  = \frac{1}{3}\int\limits_1^8 {{e^u}du} \\
 = \left. {\frac{1}{3}{e^u}} \right|_1^8 = \frac{1}{3}\left( {{e^8} - e} \right)
\end{array}\)

b) Đặt \(u = lnx \Rightarrow du = \frac{{dx}}{x}\)

\(\begin{array}{l}
\int\limits_1^3 {\frac{1}{x}} {\left( {\ln x} \right)^2}dx = \int\limits_0^{\ln 3} {{u^2}} du\\
 = \left. {\frac{{{u^3}}}{3}} \right|_0^{\ln 3} = \frac{1}{3}{\left( {\ln 3} \right)^3}
\end{array}\)

c) Đặt \(u = \sqrt {1 + {x^2}}  \Rightarrow {u^2} = 1 + {x^2} \)

\(\Rightarrow udu = xdx\)

\(\begin{array}{l}
\int\limits_0^{\sqrt 3 } x \sqrt {1 + {x^2}} dx\\
 = \int\limits_1^2 u .udu = \left. {\frac{{{u^3}}}{3}} \right|_1^2 = \frac{7}{3}
\end{array}\)

d) Đặt \(u = 3{x^3} \Rightarrow du = 9{x^2}dx \)

\(\Rightarrow {x^2}dx = \frac{1}{9}du\)

\(\begin{array}{l}
\int\limits_0^1 {{x^2}} {e^{3{x^3}}}dx = \frac{1}{9}\int\limits_0^3 {{e^u}} du\\
 = \left. {\frac{1}{9}{e^u}} \right|_0^3 = \frac{1}{9}\left( {{e^3} - 1} \right)
\end{array}\)

e) Đặt \(u = 1 +\sin x \Rightarrow du = \cos xdx\)

\(\begin{array}{l}
\int\limits_0^{\frac{\pi }{2}} {\frac{{\cos x}}{{1 + {\rm{sinx}}}}} dx = \int\limits_1^2 {\frac{{du}}{u}} \\
 = \left. {\ln |u} \right|_1^2 = \ln 2
\end{array}\)

-- Mod Toán 12