Pembahasan Soal UN SMA Tapel 2017/2018 Materi Integral

Soal 1
Diketahui $\int_{0}^{3}{\left( x^{3}+px+2\right )}dx=\frac{3}{2}$. Nilai $p$ yang memenuhi adalah ...
A. -26
B. -13
C. -3
D. 3
E. 13

Konsep yang digunakan dalam perhitungan:

• $\int{kx^{n}}dx=\frac{k}{n+1}x^{n+1}+c$
• $\int{kf(x)}dx=k\int{f(x)}dx$
• $\int{\left(f(x)\pm g(x) \right )}dx=\int{f(x)}dx\pm\int{g(x)}dx$

Pembahasan
$\frac{1}{3}x^{3}+\frac{p}{2}x^{2}+2x\left\{\begin{matrix}0\\ \\3\end{matrix}\right.=\frac{3}{2}$
$\left [ \frac{1}{3}3^{3}+\frac{p}{2}3^{2}+2.3 \right ]-\left[\frac{1}{3}0^{3}+\frac{p}{2}0^{2}+2.0 \right ]=\frac{3}{2}$
$\left[9+\frac{p}{2}.9+6\right]-[0]=\frac{3}{2}$
$\frac{9p}{2}+15=\frac{3}{2}$
$\frac{9p}{2}=\frac{3}{2}-15$
$\frac{9p}{2}=\frac{3-30}{2}$
$9p=-27$
$p=-3$
Jawaban C

Soal 2
Hasil dari $\int 2x^{2}\left(x^{3}+5 \right )^{5} dx=$ ...
A. $\frac{1}{18}\left(x^{3}+2 \right )^{6}+C$
B. $\frac{1}{9}\left(x^{3}+2 \right )^{6}+C$
C. $\frac{1}{6}\left(x^{3}+2 \right )^{6}+C$
D. $\frac{1}{3}\left(x^{3}+2 \right )^{6}+C$
E. $\frac{2}{3}\left(x^{3}+2 \right )^{6}+C$

Konsep yang digunakan dalam perhitungan:

• $\int{kx^{n}}dx=\frac{k}{n+1}x^{n+1}+c$
• $\int{kf(x)}dx=k\int{f(x)}dx$

Pembahasan
$\int 2x^{2}\left(x^{3}+5 \right )^{5} dx$
Misal: $u=x^{3}+2$
$du=3x^{2}$
$=2\int{x^{2}\left(x^{3}+2 \right )^{5}}dx$
$=2\int{\left(x^{3}+2 \right )^{5}\frac{3x^{2}}{3}}dx$
$=\frac{2}{3}\int{u^{5}}du$
$=\frac{2}{3}\left(\frac{1}{6}u^{6}+c \right )$
$=\frac{2}{3}\frac{1}{6}u^{6}+\frac{2}{3}.c$
$=\frac{2}{3}\frac{1}{6}u^{6}+C$
$=\frac{2}{18}\left(x^{3}+2 \right )^{6}+C$
$=\frac{1}{9}\left(x^{3}+2 \right )^{6}+C$
Jabawan B