Đặt: \( \begin{cases}
u = x \\
dv = \sin x \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = dx \\
v = -\cos x.
\end{cases} \)
\( \int x \sin x \, dx = -x \cos x + \int \cos x \, dx \)
\( = -x \cos x + \sin x + C. \)
page85
Đặt: \( \begin{cases}
u = x \\
dv = (1 + \sin 2x) \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = dx \\
v = x - \frac{1}{2} \cos 2x.
\end{cases} \)
\( I = x \left(x - \frac{1}{2} \sin 2x \right) - \int \left(x - \frac{1}{2} \cos 2x \right) \, dx \)
\( = x^2 - \frac{1}{2} x \sin 2x - \frac{x^2}{2} + \frac{1}{4} \sin 2x + C. \)
\( = \frac{x^2}{2} - \frac{x}{2} \sin 2x + \frac{1}{4} \sin 2x + C. \)
page86
\( I = \int x \frac{(1 - \cos 2x)}{2} \, dx. \)
Đặt: \( \begin{cases}
u = \frac{x}{2} \\
dv = (1 - \cos 2x) \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{1}{2} \, dx \\
v = x - \frac{1}{2} \sin 2x.
\end{cases} \)
\( I = \frac{x}{2} \left(x - \frac{1}{2} \sin 2x \right) - \frac{1}{2} \int \left(x - \frac{1}{2} \sin 2x \right) \, dx \)
\( = \frac{x^2}{2} - \frac{x}{4} \sin 2x - \frac{x}{2}+ \frac{1}{4} \sin 2x + C. \)
\( I = \int \frac{x}{4}(\cos 3x + 3\cos x) \)
page87
Đặt: \( \begin{cases}
u = x^2 - 2x \\
dv = (\sin x + 2\cos x) \, dx
\end{cases} \quad \Rightarrow \quad \begin{cases}
du = (2x - 2) \, dx \\
v = 2\sin x - \cos x.
\end{cases} \)
\( I = (x^2 - 2x)(2\sin x - \cos x) - \int (2x - 2)(2\sin x - \cos x) \, dx \)
Đặt: \( \begin{cases}
u = 2x - 2 \\
dv = (2\sin x - \cos x) \, dx
\end{cases} \quad \Rightarrow \quad \begin{cases}
du = 2 \, dx \\
v = -2\cos x - \sin x.
\end{cases} \)
\(I = (x^2 - 2x)(2\sin x - \cos x) - \left[-(2x - 2)(2\cos x + \sin x) + 2 \int (2\cos x + \sin x)\right] \, dx\)
\( = (x^2 - 2x)(2\sin x - \cos x) + (2x - 2)(2\cos x + \sin x) - 2(2\sin x - \cos x) + C.\)
page88
Đặt \( t = \sqrt{x} \Rightarrow x = t^2 \Rightarrow dx = 2t \, dt \).
\( \int \sin \sqrt{x} \, dx = 2 \int t \sin t \, dt. \)
Đặt:\(
\begin{cases}
u = t \\
dv = \sin t \, dt
\end{cases}
\quad \Rightarrow \quad
\begin{cases}
du = dt \\
v = -\cos t.
\end{cases}
\)
\( 2 \int t \sin t \, dt = 2 \left[-t \cos t + \int \cos t \right] . \)
\( = -2t \cos t + 2\sin t \)
\(= -2 \sqrt{x} \cos \sqrt{x} + 2 \sin \sqrt{x} + C. \)
page89