Contoh Soal Integral Fungsi Aljabar beserta Pembahasannya #3
Hai sob, pada postingan kali ini, mimin sajikan lanjutan beberapa contoh soal dan pembahasan pada materi integral fungsi aljabar (kelas 11 SMA). Cuss, langsung saja. Berikut contoh-contoh soal dan pembahasannya. Selamat belajar. Semoga bermanfaat.
Contoh soal 1
Hasil dari $\displaystyle \int (x^3-x^2-x+5)~dx$ adalah ...
Jawab:
$\begin{align} \displaystyle \int (x^3-x^2-x+5)~dx &= \frac{1}{4}x^4-\frac{1}{3}x^3-\frac{1}{2}x^2+5x+C \end{align}$
Contoh soal 2
Hasil dari $\displaystyle \int_{0}^{2} 3(x+1)(x-6)~dx$ adalah ...
Jawab:
$\begin{aligned} \displaystyle \int_{0}^{2} 3(x+1)(x-6)~dx &= \displaystyle \int_{0}^{2} 3x^2-15x-18~dx\\ &= \displaystyle x^{3} -\frac{15}{2}x^2-18x \Bigr|_{0}^{2} \\ &= \left [2^{3} -\frac{15}{2}2^2-18.2 \right ] - \left [ 0^{3} -\frac{15}{2}0^2-18.0 \right ] \\ &= 8-\frac{15}{2}.4-36 \\ &= -58 \end{aligned}$
Contoh soal 3
Nilai $\displaystyle \int_{p}^{4} (3x^2-24x+41)~dx=6$ maka nilai $p$ adalah ...
Jawab:
$\begin{aligned} \displaystyle \int_{p}^{4} (3x^2-24x+41)~dx=6\\ \displaystyle x^{3} -12x^2+41x \Bigr|_{p}^{4} &= 6 \\ \left [4^{3} -12.4^2+41.4 \right ] - \left [ p^{3} -12p^2+41.p \right ] &= 6\\ \left [36 \right ] - \left [ p^{3} -12p^2+41p \right ] &= 6\\ p^{3} -12p^2+41p&=30\\ p^{3} -12p^2+41p-30&=0\\ (p-1)(p-5)(p-6)&=0\\ p=1 \vee p=5 \vee p&=6 \end{aligned}$
Jadi, nilai p yang memenuhi adalah $p=1$
Contoh soal 4
Jika $p$ banyaknya faktor prima dari $42$ dan $q$ akar positif persamaan $3x^2-5x-2=0$, nilai dari $\displaystyle \int_{q}^{p} (5-3x)~dx=...$
Jawab:
$\bullet$ $p=$ banyaknya faktor prima dari $42$ $=\color{red}{3}$
$\bullet$ $q=$ akar positif persamaan $3x^2-5x-2=0$ $=\color{red}{2}$
sehingga didapat
$\begin{aligned} \displaystyle \int_{q}^{p} (5-3x)~dx&= \int_{2}^{3} (5-3x)~dx\\ &= \displaystyle 5x -\frac{3}{2}x^2 \Bigr|_{2}^{3} \\ &= \left [5.3 -\frac{3}{2}.3^2 \right ] - \left [ 5.2 -\frac{3}{2}.2^2 \right ] \\ &=-\frac{5}{2}\\&= -2\frac{1}{2} \end{aligned}$
Contoh soal 5
Jika $\displaystyle \int_{0}^{1} f(x)~dx=2$ dan $\displaystyle \int_{2}^{1} 2f(x)~dx=2$, maka $\displaystyle \int_{0}^{2} f(x)~dx=...$
Jawab:
Dari soal diketahui
$\begin{aligned} & \displaystyle \int_{0}^{1} f(x)~dx=2 \\ \\&\displaystyle \int_{2}^{1} 2f(x)~dx=2\\ &\displaystyle 2 \int_{2}^{1} f(x)~dx=2\\ &\displaystyle \int_{1}^{2} f(x)~dx=-1 \end{aligned}$
$\begin{aligned} \displaystyle \int_{0}^{2} f(x)~dx &= \int_{0}^{1} f(x)~dx+\int_{1}^{2} f(x)~dx\\ &= 2-1\\ &= 1\end{aligned}$
Contoh soal 6
Diketahui $\displaystyle \int_{a}^{b} \frac{1}{x-1}~dx=c$. Nilai dari $\displaystyle \int_{a}^{b} \frac{x+1}{2(x-1)}~dx=...$
Jawab:
$\begin{aligned} \displaystyle \int_{a}^{b} \frac{x+1}{2(x-1)}~dx&= \frac{1}{2}( \int_{a}^{b} \frac{x+1}{(x-1)})~dx\\ &= \frac{1}{2}( \int_{a}^{b} \frac{x}{(x-1)}~dx+\int_{a}^{b} \frac{1}{(x-1)})~dx\\ &= \frac{1}{2}( \int_{a}^{b} \frac{x-1+1}{(x-1)}~dx+\int_{a}^{b} \frac{1}{(x-1)})~dx \\ &=\frac{1}{2}( \int_{a}^{b} \frac{x-1}{(x-1)}~dx+ \int_{a}^{b} \frac{1}{(x-1)}~dx+\int_{a}^{b} \frac{1}{(x-1)})~dx\\ &=\frac{1}{2}( \displaystyle x \Bigr|_{a}^{b}+c+c)\\ &= \frac{1}{2}( b-a+2c)\\ &=\frac{1}{2}(b-a)+c \end{aligned}$
Contoh soal 7
Jika $\displaystyle \int_{1}^{4} f(x)~dx=6$ maka $\displaystyle \int_{1}^{4} f(5-x)~dx=...$
Jawab:
$\begin{aligned} \displaystyle \int_{1}^{4} f(x)~dx&=6\\ \displaystyle F(x) \Bigr|_{1}^{4}&=6\\ F(4)-F(1)&=6\\ \\ \displaystyle \int_{1}^{4} f(5-x)~dx&= \displaystyle F(5-x) \Bigr|_{1}^{4} \\ &= F(5-4)-F(5-1)\\ &= F(1)-F(4)\\ &= -(F(4)-F(1))\\ &= -6 \end{aligned}$
Demikianlah beberapa contoh soal dan pembahasan pada materi integral fungsi aljabar. Semoga bermanfaat.
Referensi:
Tim BBM. 2015. Big Book Matematika SMA Kelas 1, 2, dan 3. Jakarta: Cmedia.
