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:
\(\left [ -\frac{1}{5} \cos 5x-(-\cos x)\right ]_{0}^{\frac{\pi }{2}} \)
\(\left [ -\frac{1}{5}\cos 5x+\cos x \right ]_{0}^{\frac{\pi }{2}}\)
\(\left ( -\frac{1}{5} \cos 5*\frac{\pi }{2}+\cos \frac{\pi }{2}\right )-\left ( -\frac{1}{5}\cos 5*0+\cos 0 \right )\)
\(\left ( -\frac{1}{5}*0+0 \right )-\left ( -\frac{1}{5}*1+1 \right )\)
\(0+\frac{1}{5}-1\)
\(-\frac{4}{5}\)
