Tìm \(H=\int \frac{x^{2} d x}{(x \sin x+\cos x)^{2}} ?\)

\(\text { Ta có: } H=\int \frac{x^{2}}{(x \sin x+\cos x)^{2}} d x=\int \frac{x \cos x}{(x \sin x+\cos x)^{2}} \cdot \frac{x}{\cos x} d x\)

\(\begin{array}{l}
\text { Đặt }\left\{\begin{array}{l}
u=\frac{x}{\cos x} \\
d v=\frac{x \cos x}{(x \sin x+\cos x)^{2}} d x=\frac{d(x \sin x+\cos x)}{(x \sin x+\cos x)^{2}}
\end{array} \Rightarrow\left\{\begin{array}{l}
d u=\frac{x \sin x+\cos x}{\cos ^{2} x} d x \\
v=-\frac{1}{x \sin x+\cos x}
\end{array}\right.\right.
\end{array}\)

\(\Rightarrow H=-\frac{x}{\cos x} \cdot \frac{1}{x \sin x+\cos x}+\int \frac{1}{\cos ^{2} x} d x\\
=\frac{-x}{\cos x(x \sin x+\cos x)}+\tan x+C\)