Bài tập 9 trang 146 SGK Toán 12 NC

a)

Đặt

\(\left\{ \matrix{
u = {x^2} \hfill \cr 
dv = \cos 2xdx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = 2xdx \hfill \cr 
v = {1 \over 2}\sin 2x \hfill \cr} \right.\) 

Do đó \(\int {{x^2}\cos 2xdx}\) \( = {1 \over 2}{x^2}\sin 2x - \int {x\sin 2xdx\,\,\,\left( 1 \right)} \) 

Tính \(\int {x\sin 2xdx} \) 

Đặt 

\(\left\{ \matrix{
u = x \hfill \cr 
dv = \sin 2xdx \hfill \cr} \right. \Rightarrow \left\{ \matrix{
du = dx \hfill \cr 
v = - {1 \over 2}\cos 2x \hfill \cr} \right.\)

\( \Rightarrow \int {x\sin 2xdx }\) \(=  - {1 \over 2}x\cos 2x + {1 \over 2}\int {\cos 2xdx }\) \( =  - \dfrac{1}{2}x\cos 2x + \dfrac{1}{2}.\dfrac{{ - \cos 2x}}{2} + {C_1}\)  \( =  - {1 \over 2}x\cos 2x - {1 \over 4}\sin 2x + C_1 \)

Thay vào (1) ta được \(\int {{x^2}\cos 2xdx }\)

\( = \dfrac{1}{2}{x^2}\sin 2x \) \(- \left( { - \dfrac{1}{2}x\cos 2x - \dfrac{1}{4}\sin 2x + {C_1}} \right)\) 

\(= {1 \over 2}{x^2}\sin 2x + {1 \over 2}x\cos 2x + {1 \over 4}\sin 2x + C \)

 

b) Đặt

\(\begin{array}{*{20}{l}}
{\left\{ {\begin{array}{*{20}{l}}
{u = \ln x}\\
{dv = \sqrt x dx}
\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}
{du = \frac{{dx}}{x}}\\
{v = \frac{2}{3}{x^{\frac{3}{2}}}}
\end{array}} \right.}\\
{ \Rightarrow \int {\sqrt x } \ln xdx = \frac{2}{3}{x^{\frac{3}{2}}}\ln x - \frac{2}{3}\int {{x^{\frac{1}{2}}}} dx}\\
\begin{array}{l}
 = \frac{2}{3}{x^{\frac{3}{2}}}\ln x - \frac{2}{3}.\frac{2}{3}{x^{\frac{3}{2}}} + C\\
 = \frac{2}{3}\sqrt {{x^3}} \ln x - \frac{4}{9}\sqrt {{x^3}}  + C
\end{array}
\end{array}\)

c) Đặt \(u = sinx \Rightarrow du = cosxdx\)

\(\begin{array}{l}
 \Rightarrow \int {{{\sin }^4}} x\cos xdx = \int {{u^4}} du\\
 = \frac{{{u^5}}}{5} + C = \frac{1}{5}{\sin ^5}x + C.
\end{array}\)

d) Đặt \(u = {x^2} \Rightarrow du = 2xdx \)

\(\Rightarrow xdx = \frac{1}{2}du\)

\(\begin{array}{l}
 \Rightarrow \int x \cos \left( {{x^2}} \right)dx = \frac{1}{2}\int {\cos } udu\\
 = \frac{1}{2}\sin u + C = \frac{1}{2}{\rm{sin}}{{\rm{x}}^2} + C
\end{array}\)

-- Mod Toán 12 HỌC247