Tìm nguyên hàm của các hàm số sau:
\(\begin{array}{l}
a)f\left( x \right) = {x^2}\cos 2x\\
b)f\left( x \right) = \sqrt x \ln x\\
c)f\left( x \right) = {\sin ^4}x\cos x\\
d)f(x) = x\cos ({x^2})
\end{array}\)
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