Haii adik-adik...
Selamat datang di halaman contoh soal dan pembahasan "Integral". Di halaman ini akan membahas tentang contoh soal dan pembahasan lengkap mengenai Integral.
Untuk lebih jelasnya mari kita lihat contoh soal dan pembahasan dibawah ini!
21. hasil dari \(\int\limits \cos ^3 x dx\)
a) \(\sin x-\frac{1}{3}\sin ^3 x +c\)
b) \(\frac{1}{4}\cos ^4 x+c\)
c) \(3\cos ^2x\sin x+c\)
d) \(\frac{1}{3}\sin ^3x-\sin x+c\)
e) \(\sin x-3\sin ^3x+c\)
Pembahasan:
\(\int \cos ^3 x dx\)
\(\int \cos x\cos ^2x dx\)
\(\int \cos x(1-\sin ^2x)dx \)
\(\int \cos x-\int \sin ^2x\cos xdx \)
\(\sin x-\int u^2\cos x \frac{du}{\cos x} \)
\(\sin x-\frac{1}{3}u^3+c\)
\(\sin x-\frac{1}{3}\sin ^3x+c\)
22. nilai dari \(\int_{0}^{2}\limits x^2(x+2)dx\)=
a) \(6\)
b) \(6\frac{1}{3}\)
c) \(6\frac{2}{3}\)
d) \(9\frac{1}{3}\)
e) \(20\)
Pembahasan:
\(\int_{0}^{2}x^2(x+2)dx\)
\(\int_{0}^{2}x^3+2x^2 dx \)
\(\frac{1}{4}x^2+\frac{2}{3}x^3\)
\(\left (\frac{1}{4}*2^4+\frac{2}{3}*2^3 \right )-\left ( \frac{1}{4}*0^4+\frac{2}{3}*0^3 \right )\)
\(4+\frac{16}{3}\)
\(\frac{12+6}{3}\)
\(9\frac{1}{3}\)
23. hasil dari \(\int\limits \sin 3x\cos 2x dx \)=
a) \(-\frac{1}{5}\cos 5x+\frac{1}{2}\cos x+c\)
b) \(-\frac{1}{10}\cos 5x-\frac{1}{2}\cos x+c\)
c) \(-\sin \frac{1}{2}x-5\sin \frac{5}{2}x+c\)
d) \(\frac{1}{25}\sin 5x+\sin x+c\)
e) \(\cos 5x-\cos x+c\)
Pembahasan:
\(\int \sin 3x\cos 2xdx\)
\(\frac{1}{2}\int \sin (3x+2x)+\sin (3x-2x)\)
\(\frac{1}{2}\int \sin 5x+\sin x dx\)
\(\frac{1}{2}\left [ -\frac{1}{5}\cos 5x-\cos x \right ] +c \)
\(-\frac{1}{10}\cos 5x-\frac{1}{2}\cos x+c\)
24. hasil \(\int_{1}^{3}\limits \left ( x^2+\frac{1}{6} \right ) dx\)
a) \(9\frac{1}{3}\)
b) \(9\)
c) \(8\)
d) \(\frac{10}{3}\)
e) \(3\)
Pembahasan:
\(\int_{1}^{3}\left ( x^2+\frac{1}{6} \right ) dx \)
\(\frac{1}{3}x^3+\frac{1}{6}x \)
\(\left (\frac{1}{3}*3^3+\frac{1}{6}*3 \right )-\left ( \frac{1}{3}*1^3+\frac{1}{6*1} \right )\)
\(\left ( 9+\frac{1}{2} \right )-\left ( \frac{1}{3}+\frac{1}{6} \right )\)
\(\left ( 9+\frac{1}{2} \right )-\frac{1}{2}\)
\(9\)
25. Diketahui \(\int_{1}^{p}\limits3x\left( x+\frac{2}{3} \right)dx=78\). Nilai \((-2p)\) adalah
a) \(8\)
b) \(4\)
c) \(0\)
d) \(-4\)
e) \(-8\)
Pembahasan:
\(\int_{1}^{p}3x\left( x+\frac{ 2}{ 3} \right)dx=78\)
\(\int_{1}^{p}3x^2+2xdx=78x^3+x^2|_{1}^{p}=78\)
\(\left( p^3+p^2 \right)-\left( 1+1 \right)=78\)
\(p^3+p^2-80=0\rightarrow p=4\)
jadi, nilai \(-2p=-8\)
26. hasil \(\int\limits 6x\sqrt{3x^2+5}dx\)
a) \(\frac{2}{3}(6x^2+5)\sqrt{6x^2+5}+c\)
b) \(\frac{2}{3}(3x^2+5)\sqrt{3x^2+5}+c\)
c) \(\frac{2}{3}(x^2+5)\sqrt{x^2+5}+c\)
d) \(\frac{3}{2}(x^2+5)\sqrt{x^2+5}+c\)
e) \(\frac{3}{2}(3x^2+5)\sqrt{3x^2+5}+c\)
Pembahasan:
\(\int 6x\sqrt{3x^2+5}dx \)
\(\int 6x*u^\frac{1}{2}\frac{du}{6x}\)
\(\int u^\frac{1}{2}du=\frac{1}{\frac{1}{2}+1}u^\frac{3}{2}+c \)
\(\frac{2}{3}u^\frac{3}{2}+c \)
\(\frac{2}{3}(3x^2+5)\sqrt{3x^2+5}+c\)
27. \(\int_{0}^{2}\limits (3x^2-3x+7)dx\)=
a) \(6\)
b) \(10\)
c) \(13\)
d) \(16\)
e) \(22\)
Pembahasan:
\(\int_{0}^{2}(3x^2-3x+7)dx\)
\((x^3-\frac{3}{2}x^2+7x)\)
\(\left ( 2^3-\frac{3}{2}*2^3+7*2 \right )-\left ( 0^3-\frac{3}{2}*0^2+7*0 \right )\)
\(16\)
28. nilai dari \(\int_{1}^{3}\limits(3x^2+2x-1)dx\)
a) \(12\)
b) \(28\)
c) \(32\)
d) \(33\)
e) \(34\)
Pembahasan:
\(\int_{1}^{3}(3x^2+2x-1)dx\)
\(\left [ x^3+x^2-x \right ]_{1}^{3}\)
\(\left ( 3^3+3^2-3 \right )-\left ( 1^3+1^2-1 \right )\)
\(33-1\)
\(32\)
29. hasil dari \(\int_{0}^{2}\limits 3(x+1)(x-6)dx\)
a) \(-58\)
b) \(-56\)
c) \(-28\)
d) \(-16\)
e) \(-14\)
\(\int_{0}^{2}3(x+1)(x-6)dx\)
\(\int_{0}^{2}(3x^2-15x-18)dx \)
\(\left [ x^3-\frac{15}{2}x^2-18x \right ]_{0}^{2} \)
\((2^3-\frac{15}{2}*2^2-18*2)-\left ( 0^3-\frac{15}{2}*0^2-18*0 \right )\)
\(8-30-36-(0)\)
\(-58\)
30. hasil dari \(\int_{0}^{\frac{\pi }{2}}\limits (\sin 5x-\sin x) dx\)
a) \(-\frac{4}{5}\)
b) \(-\frac{1}{5}\)
c) \(-\frac{1}{2}\)
d) \(1\)
e) \(\frac{4}{5}\)
Pembahasan:
\(\left [ -\frac{1}{5} \cos 5x-(-\cos x)\right ]_{0}^{\frac{\pi }{2}} \)
\(\left [ -\frac{1}{5}\cos 5x+\cos x \right ]_{0}^{\frac{\pi }{2}}\)
\(\left ( -\frac{1}{5} \cos 5*\frac{\pi }{2}+\cos \frac{\pi }{2}\right )-\left ( -\frac{1}{5}\cos 5*0+\cos 0 \right )\)
\(\left ( -\frac{1}{5}*0+0 \right )-\left ( -\frac{1}{5}*1+1 \right )\)
\(0+\frac{1}{5}-1\)
\(-\frac{4}{5}\)
