Cho $A=\int \dfrac{\cos ~x}{\sin x+\cos ~x}dx,B=\int \dfrac{\sin...

Phương pháp:
- Phân tích $\sin x+3\cos x=M(\operatorname{s}\text{in}x+\cos x)+N(\cos x-\operatorname{s}\text{in} x),$ tìm M,N.
- Sử dụng các nguyên hàm cơ bản: $\int dx=x+C, \int \dfrac{dx}{x}=lnx+C.~$
Cách giải:
K=3A+B
$K=3\int \dfrac{\cos x}{\operatorname{s}\text{in} x+\cos x}dx+\int \dfrac{\sin x}{\cos x+\operatorname{s}\text{in} x}~dx$
$K=\int \dfrac{s\text{in} x+3\cos x}{\operatorname{s}\text{in} x+\cos x}~dx$
Giả sử
$\sin x+3\cos x=M\left( \sin x+\cos x \right)+N\left( \cos x-~\sin ~x \right)$
$=\left( M-N \right)\sin x+\left( M+~N \right)~\cos ~x~$
Đồng nhất hệ số ta có:$\left\{ \begin{aligned}
& M-N=1 \\
& M+N~=3~ \\
\end{aligned} \right.~\Leftrightarrow \left\{ \begin{aligned}
& M=2 \\
& N=2 \\
\end{aligned} \right.~~$
$\Rightarrow \sin x+3\cos x=2\left( \sin x+\cos x \right)+\left( \cos x-\sin x \right).~$
Khi đó ta có:
$K=\int{\dfrac{2\left( \sin x+\cos x \right)}{\sin x+\cos x}}dx+\int{\dfrac{\left( \cos x-\sin x~ \right)}{\sin x+\cos x}}dx~$
$K=\int{2dx}+\int{\dfrac{\left( \sin x+\cos x \right)'}{\sin x+\cos x} dx}~$
$K=2x+\ln \left| \operatorname{s}\text{in} x+\cos x \right|~+C~$