Integral Tertentu

Jika f kontinu pada interval [a, b] dan F adalah antiturunan dari f pada interval tersebut, maka :

\int_{a}^{b} f (x) d x = {[F (x)]}_{a}^{b} = F (b) - F (a)

$\mathrm{\int_{a}^{b}f(x)\;dx=\left [ F(x) \right ]_{a}^{b}=F(b)-F(a)}$
Contoh :
1.

$\mathrm{\int_{1}^{3}(2x-1)\;dx}$

$\mathrm{=\left [ x^{2}-x \right ]_{1}^{3}}$

$\mathrm{=\left ( 3^{2}-3 \right )-\left ( 1^{2}-1 \right )}$

$=6-0$

$=6$

2.

$\mathrm{\int_{0}^{1}\left ( x^{2}-3x \right )\;dx}$

$\mathrm{=\left [ \frac{1}{3}x^{3}-\frac{3}{2}x^{2} \right ]_{0}^{1}}$

$\mathrm{=\left (\frac{1}{3}(1)^{3}-\frac{3}{2}(1)^{2} \right )-\left (\frac{1}{3}(0)^{3}-\frac{3}{2}(0)^{2} \right )}$

$=\frac{1}{3}-\frac{3}{2}$

$=-\frac{7}{6}$

Sifat-Sifat Integral Tertentu

$\mathrm{\int_{a}^{b}k\:f(x)\;dx=k\int_{a}^{b}f(x)\:dx}$

2.

$\mathrm{\int_{a}^{b}\left [f(x)\pm g(x) \right ]\;dx=\int_{a}^{b}f(x)\:dx\pm \int_{a}^{b}g(x)\:dx}$

3.

$\mathrm{\int_{a}^{a}f(x)\;dx=0}$

4.

$\mathrm{\int_{a}^{b}f(x)\;dx=-\int_{b}^{a}f(x)\;dx}$

5.

$\mathrm{\int_{a}^{c}f(x)\;dx=\int_{a}^{b}f(x)\;dx+\int_{b}^{c}f(x)\;dx}$
dengan : a < b < c

Berikut contoh-contoh latihan soal integral tertentu :

Contoh 1
Jika

$\mathrm{f(x)=3x^{2}-4x}$ dan

$\mathrm{g(x)=2x+1}$ , tentukan nilai dari :

a.

$\mathrm{\int_{1}^{3}f(x)\:dx}$

Jawab :

$\mathrm{\Rightarrow \int_{1}^{3}\left ( 3x^{2}-4x \right )\:dx}$

$\mathrm{= \left [x^{3}-2x^{2} \right ]_{1}^{3}}$

$\mathrm{= \left ((3)^{3}-2(3)^{2} \right )-\left ((1)^{3}-2(1)^{2} \right )}$

$=9-(-1)$

$\mathrm{=10}$

b.

$\mathrm{\int_{-2}^{1}g(x)\:dx}$

Jawab :

$\mathrm{\Rightarrow\int_{-2}^{1}(2x+1)\:dx}$

$\mathrm{=\left [x^{2}+x \right ]_{-2}^{1}}$

$\mathrm{=\left ( 1^{2}+1 \right )-\left ( (-2)^{2}+(-2) \right )}$

$=2-2$

$=0$

c.

$\mathrm{\int_{0}^{1}\left (f(x)-g(x) \right )\:dx}$

Jawab :

$\mathrm{\Rightarrow \int_{0}^{1}\left (\left ( 3x^{2}-4x \right )-(2x+1) \right )\:dx}$

$\mathrm{\Rightarrow \int_{0}^{1}\left (3x^{2}-6x-1 \right )\:dx}$

$\mathrm{=\left [x^{3}-3x^{2}-x \right ]_{0}^{1}}$

$\mathrm{=\left (1^{3}-3(1)^{2}-1 \right )-\left ( 0^{3}-3(0)^{2}-0 \right )}$

$=-3-0$

$=-3$

Contoh 2
Tentukan nilai a dari pengintegralan berikut !

a.

$\mathrm{\int_{a}^{a+1}\left ( 2x+1 \right )\;dx=4}$

Jawab :

$\mathrm{\Rightarrow \left [ x^{2}+x \right ]_{a}^{a+1}=4}$

$\mathrm{\Rightarrow \left [ \left ( a+1 \right )^{2}+\left ( a+1 \right ) \right ]-\left [ a^{2}+a \right ]=4}$

$\mathrm{\Rightarrow a^{2}+2a+1+a+1-a^{2}-a=4}$

$\mathrm{\Rightarrow 2a+2=4}$

$\mathrm{\Rightarrow 2a=2}$

$\mathrm{\Rightarrow a=1}$

b.

$\mathrm{\int_{1}^{a}\left ( 2x-2 \right )\;dx=2a+1\;\;;\,a>0}$

Jawab :

$\mathrm{\Rightarrow\left [ x^{2}-2x \right ]_{1}^{a}=2a+1}$

$\mathrm{\Rightarrow \left ( a^{2}-2a \right )-\left ( 1^{2}-2.1 \right )=2a+1}$

$\mathrm{\Rightarrow a^{2}-2a+1=2a+1}$

$\mathrm{\Rightarrow a^{2}-4a=0}$

$\mathrm{\Rightarrow a\left ( a-4 \right )=0}$

$\mathrm{\Rightarrow a=0\;atau\;a=4}$
Karena a > 0, maka

$\mathrm{a=4}$

c.

$\mathrm{\int_{a}^{1}\left ( 3x^{2}+2x \right )\;dx=-10\;\;;\,a\in R}$

Jawab ;

$\mathrm{\Rightarrow \left [ x^{3}+x^{2} \right ]_{a}^{1}=-10}$

$\mathrm{\Rightarrow \left (1^{3}+1^{2} \right )-\left ( a^{3}+a^{2} \right )=-10}$

$\mathrm{\Rightarrow 2-a^{3}-a^{2}=-10}$

$\mathrm{\Rightarrow a^{3}+a^{2}-12=0}$

Nilai a yang mungkin adalah faktor dari 12, yaitu : ±1, ±2, ±3, ±4, ±6, ±12.
Nilai a yang memenuhi adalah a = 2, karena :

$\mathrm{\Rightarrow 2^{3}+2^{2}-12=0}$

Contoh 3
Nilai dari

$\mathrm{\int_{0}^{\frac{\pi }{2}}\left (2\,cos\:2x-sin\,x \right )\;dx}$ adalah...

Jawab :

$\mathrm{\int_{0}^{\frac{\pi }{2}}\left (2\,cos\:2x-sin\,x \right )\;dx}$

$\mathrm{=\left [ 2\,\frac{1}{2}sin\,2x-(-cos\,x) \right ]_{0}^{\frac{\pi }{2}}}$

$\mathrm{=\left [ sin\,2x+cos\,x \right ]_{0}^{\frac{\pi }{2}}}$

$\mathrm{=\left ( sin\,2\left ( \frac{\pi }{2} \right )+cos\,\left ( \frac{\pi }{2} \right ) \right )-\left ( sin\,2(0)+cos\,0 \right )}$

$\mathrm{=\left ( sin\,\pi+cos\left ( \frac{\pi }{2} \right ) \right )-\left ( sin\,0+cos\,0 \right )}$

$=\left ( 0+0 \right )-\left ( 0+1 \right )$

$=-1$