Pembahasan Ujian PAI: A50 – No. 30 – Juni 2016

1. \(f\left( x \right) = \frac{1}{{{{\left( {1 + x} \right)}^3}}}{\rm{,}}\) untuk \(x \ge 0\)
2. \(f\left( x \right) = \frac{1}{{{{\left( {1 + x} \right)}^2}}}{\rm{,}}\) untuk \(x \ge 0\)
3. \(f\left( x \right) = \left( {2x – 1} \right){e^{ – x}}{\rm{,}}\) untuk \(x \ge 0\)

1. \(f\left( x \right) \ge 0,{\rm{ }}\)  untuk setiap \({\rm{ }}x \in R\)
2. \(\int_{ – \infty }^\infty {f\left( x \right)dx} = 1\)

\(\int\limits_0^\infty {\frac{1}{{{{\left( {1 + x} \right)}^3}}}dx} = \int\limits_1^\infty {\frac{1}{{{u^3}}}du} \)
\(= \left. { – \frac{1}{{2{u^2}}}} \right|_1^\infty \)
\(= \frac{1}{2}\) (Salah)

\(\int\limits_0^\infty {\frac{1}{{{{\left( {1 + x} \right)}^2}}}dx} = \int\limits_1^\infty {\frac{1}{{{u^2}}}du} \)
\(= \left. { – \frac{1}{u}} \right|_1^\infty \)
\(= \frac{1}{1}\) (Benar)

\(\int\limits_0^\infty {\left( {2x – 1} \right){e^{ – x}}dx} = \int\limits_0^{ – \infty } {{e^u}\left( {2u + 1} \right)du} {\rm{ }}\)  misal \(u = – x\)  maka \(du = – dx\)
\(= \int\limits_1^0 {\left( {2\ln \left( v \right) + 1} \right)dv} \)  misal \(v = {e^u}\)  maka  \(dv = {e^u}du\)
\(= 2\int\limits_1^0 {\ln \left( v \right)dv} + \int\limits_1^0 {1dv} \)
nilai  \(\int\limits_1^0 {\ln \left( v \right)dv} \)  tidak bisa ditentukan (Salah)