3.7.4 Pengamiran, SPM Praktis (Kertas 1)

Soalan 9:

Diberi y = \frac{5 x}{x^{2} + 1} dan \frac{d y}{d x} = g (x), cari nilai bagi \int_{0}^{3} 2 g (x) d x .

Penyelesaian:

\begin{array}{l} Memandangkan \frac{d y}{d x} = g (x), maka y = \int g (x) d x \\ \int_{0}^{3} 2 g (x) d x = 2 \int_{0}^{3} g (x) d x \\ = 2 {[y]}_{0}^{3} \\ = 2 {[\frac{5 x}{x^{2} + 1}]}_{0}^{3} \\ = 2 [\frac{5 (3)}{3^{2} + 1} - 0] \\ = 2 (\frac{15}{10}) \\ = 3 \end{array}

Soalan 10:

Cari \int_{5}^{k} (x + 1) d x, dalam sebutan k .

Penyelesaian:

\begin{array}{l} {\int_{5}^{k} (x + 1) d x = [\frac{x^{2}}{2} + x]}_{5}^{k} \\ = (\frac{k^{2}}{2} + k) - (\frac{5^{2}}{2} + 5) \\ = \frac{k^{2} + 2 k}{2} - \frac{35}{2} \\ = \frac{k^{2} + 2 k - 35}{2} \end{array}

Soalan 11:

\begin{array}{l} Diberi = \int_{2}^{5} g (x) d x = - 2 . Cari \\ (a) nilai bagi \int_{5}^{2} g (x) d x, \\ (b) nilai bagi m jika \int_{2}^{5} [g (x) + m (x)] d x = 19 \end{array}

Penyelesaian:

\begin{array}{l} (a) \int_{5}^{2} g (x) d x = - \int_{2}^{5} g (x) d x \\ = - (- 2) \\ = 2 \end{array}

\begin{array}{l} (b) \int_{2}^{5} [g (x) + m (x)] d x = 19 \\ \int_{2}^{5} g (x) d x + m \int_{2}^{5} x d x = 19 \\ - 2 + m {[\frac{x^{2}}{2}]}_{2}^{5} = 19 \\ \frac{m}{2} {[x^{2}]}_{2}^{5} = 21 \\ \frac{m}{2} [25 - 4] = 21 \\ 21 m = 42 \\ m = 2 \end{array}

Soalan 12:

\begin{array}{l} a) Cari nilai bagi \int_{- 1}^{1} {(3 x + 1)}^{3} d x . \\ (b) Nilaikan \int_{3}^{4} \frac{1}{\sqrt{2 x - 4}} d x . \end{array}

Penyelesaian:

\begin{array}{l} a) {\int_{- 1}^{1} {(3 x + 1)}^{3} d x = [\frac{{(3 x + 1)}^{4}}{4 (3)}]}_{- 1}^{1} \\ = {[\frac{{(3 x + 1)}^{4}}{12}]}_{- 1}^{1} \\ = \frac{1}{12} [4^{4} - {(- 2)}^{4}] \\ = \frac{1}{12} (256 - 16) \\ = 20 \end{array}

\begin{array}{l} (b) \int_{3}^{4} \frac{1}{\sqrt{2 x - 4}} d x = \int_{3}^{4} \frac{1}{{(2 x - 4)}^{\frac{1}{2}}} d x \\ = \int_{3}^{4} {(2 x - 4)}^{- \frac{1}{2}} d x \\ = {[\frac{{(2 x - 4)}^{- \frac{1}{2} + 1}}{\frac{1}{2} (2)}]}_{3}^{4} \\ = {[\sqrt{2 x - 4}]}_{3}^{4} \\ = [\sqrt{2 (4) - 4} - \sqrt{2 (3) - 4}] \\ = 2 - \sqrt{2} \end{array}