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UTS 1 Matematika IIB ITB 2017 (Bagian B nomor 1)

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Halo kembali lagi di seri soal Kalkulus ITB ! 


Diketahui fungsi  \(f(x) = \frac{ln\,x }{x^2}\)


(a). Tentukan  \(\int f(x) \, \: dx\)

(b). Periksa apakah \(\int_{t}^{\infty } f(x)\, dx\)  konvergen atau divergen

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$\int f(x)\,dx=\int\frac{\ln x}{x^2}\,dx$. Misalkan $u=\ln x$ dan $dv=\frac{dx}{x^2}$. Maka dengan integral parsial, \begin{align*}\int u\,dv&=uv-\int v\,du\\\int\frac{\ln x}{x^2}\,dx&=\ln x\cdot\left(-\frac{1}{x}\right)-\int\left(-\frac{1}{x}\right)\cdot\frac{1}{x}\,dx\\&=-\frac{\ln x}{x}+\int\frac{\,dx}{x^2}\\&=\boxed{-\frac{\ln x}{x}-\frac{1}{x}+C}.\end{align*}


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