Question

Bagaimana cara menyelesaikan integral ini dengan metode parsial?
[tex]\displaystyle \int \left ( \frac{x}{x\sin x+\cos x} \right )^2dx[/tex]

Answer 1

Verified answer

Jawaban:

[tex]\displaystyle \int \left ( \frac{x}{x\sin x+\cos x} \right )^2dx=\boxed{-\frac{x~\sec(x)}{x~\sin(x)+\cos(x)}+\tan(x)+C},atau\\\boxed{\frac{(-x\cos(x)+\sin(x))}{(x\sin(x)+\cos(x))}+C}[/tex]

Penjelasan dengan langkah-langkah:

Integrate by parts (geser kekanan)

• [tex]\int~uv'~dx~\Rightarrow~uv-\int~u'v~dx[/tex]

Pengerjaan ada difoto 1 dan 2

Bentuk Lain

[tex]\begin{aligned}\displaystyle~-\frac{x}{\cos(x)(x\sin(x)+\cos(x))}+\tan(x)&=-\frac{x}{\cos(x)(x\sin(x)+\cos(x))}+\frac{\sin(x)}{\cos(x)}\\&=\frac{-x+\sin(x)(x\sin(x)+\cos(x))}{\cos(x)(x\sin(x)+\cos(x))}\\&=\frac{-x+x\sin^2(x)+\sin(x)\cos(x)}{\cos(x)(x\sin(x)+\cos(x))}\\&=\frac{-x(1-\sin^2(x))+\sin(x)\cos(x)}{\cos(x)(x\sin(x)+\cos(x))}\\&\cdots1-\sin^2(x)~=\cos^2(x)\\&=\frac{-x(\cos^2(x))+\sin(x)\cos(x)}{\cos(x)(x\sin(x)+\cos(x))}\\&=\frac{\cos(x)(-x\cos(x)+\sin(x))}{\cos(x)(x\sin(x)+\cos(x))}\\&=\boxed{\frac{(-x\cos(x)+\sin(x))}{(x\sin(x)+\cos(x))}+C}\end{aligned}[/tex]