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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!
31. Tentukan nilai dari \(\int_{1}^{3}\limits 2x(3x+4)dx=\)
a) \(88\)
b) \(84\)
c) \(56\)
d) \(48\)
e) \(46\)
Pembahasan:
\(\int_{-1}^{3}2x\left ( 3x+4 \right )dx\)
\(\int_{-1}^{3}\left ( 6x^{2}+8x \right )dx\)
\(\left [ 2x^{3}+4x^{2} \right ]_{-1}^{3}\)
\(\left [ 2\times 27+4\times 9 \right ]-\left [ 2\left ( -1 \right )+4\times 1 \right ]\)
\(90-2\)
\(88\)
32. hasil dari \(\int\limits \sin \left ( \frac{1}{2}x-\pi \right )\cos \left ( \frac{1}{2}x-\pi \right )dx\)
a) \(-2\cos \left ( x-2\pi \right )+ C\)
b) \(-\frac{1}{2}\cos \left ( x-2\pi \right )+ C\)
c) \(\frac{1}{2}\cos \left ( x-2\pi \right )+ C\)
d) \(\cos \left ( x-2\pi \right )+ C\)
e) \(2\cos \left ( x-2\pi \right )+ C\)
Pembahasan:
\(\frac{1}{2}\int \sin 2\left ( \frac{1}{2}x-\pi \right )dx\)
\(\frac{1}{2}\int \sin \left ( x-2\pi \right )dx\)
\(\frac{1}{2}\left [ -\cos \left ( x-2\pi \right ) \right ]+C\)
\(-\frac{1}{2}\cos \left ( x-2\pi \right )+C\)
33. \(\int_{0}^{\frac{1}{2}\pi }\limits \left ( 2\sin x\cos x \right )dx\)
a) \(-1\)
b) \(-\frac{1}{2}\sqrt{3}\)
c) \(\frac{1}{2}\)
d) \(\frac{1}{2}\sqrt{3}\)
e) \(1\)
Pembahasan:
\(\int_{0}^{\frac{1}{2}\pi }\sin 2x\)
\(\left [ -\frac{1}{2}\cos 2x \right ]_{0}^{\frac{1}{2}\pi }\)
\(\left [ -\frac{1}{2}\cos 2\left ( 90^{\circ} \right ) \right ]-\left [ -\frac{1}{2}\cos 0^{\circ} \right ]\)
\(\left ( -\frac{1}{2}\cos 180^{\circ} \right )-\left ( -\frac{1}{2}\cos 0^{\circ} \right )\)
\(\left [ -\frac{1}{2}. -1 \right ]-\left [ -\frac{1}{2}. 1 \right ]\)
\(\frac{1}{2}+\frac{1}{2}\)
\(1\)
34. Hasil dari \(\int\limits \left ( 6x^{2}-4x \right )\sqrt{\left ( x^{3}-x^{2}-1 \right )}dx\)
a) \(\frac{2}{3}\sqrt[3]{\left ( x^{3}-x^{2}-1^{2} \right )}+C\)
b) \(\frac{2}{3}\sqrt{\left ( x^{3}-x^{2}-1 \right )^{3}}+C\)
c) \(\frac{4}{3}\sqrt{\left ( x^{3}-x^{2}-1 \right )^{3}}+C\)
d) \(\frac{4}{3}\sqrt[3]{\left ( x^{3}-x^{2}-1^{2} \right )}+C\)
e) \(\frac{2}{3}\sqrt{\left ( x^{3}-x^{2}-1 \right )}+C\)
Pembahasan:
\(\int \left ( 6x^{2}-4x \right )\sqrt{\left ( x^{3}-x^{2-1} \right )}\)
\(\int \sqrt{u} 2 du\)
\(2\int u^{\frac{1}{2}} du\)
\(2\left ( \frac{1}{\frac{1}{2}+1} \right )u^{\frac{1}{2}+1}+C\)
\(2 . \frac{1}{\frac{3}{2}}.u^{\frac{3}{2}}+C\)
\(2.\frac{2}{3}u^{\frac{3}{2}}+C\)
\(\frac{4}{3}\sqrt{\left ( x^{3}-x^{2}-1 \right )^{3}}+C\)
35. Hasil dari \(\int\limits \sin 3x\cos x dx\)
a) \(-\frac{1}{8}\cos 4x-\frac{1}{4}\cos 2x+C\)
b) \(\frac{1}{8}\cos 4x+\frac{1}{4}\cos 2x+C\)
c) \(-\frac{1}{4}\cos 4x-\frac{1}{2}\cos 2x+C\)
d) \(\frac{1}{8}\cos 4x+\frac{1}{4}\cos 2x+C\)
e) \(-4\cos 4x-2\cos 2x+C\)
Pembahasan:
\(\int \frac{1}{2}\left \{ \sin \left ( 3x+x \right ) +\sin \left ( 3x-x \right )\right \}\)
\(\frac{1}{2}\int \left ( \sin 4x+\sin 2x \right )dx\)
\(\frac{1}{2}\left ( -\frac{1}{4}\cos 4x-\frac{1}{2}\cos 2x \right )+C\)
\(-\frac{1}{8}\cos 4x-\frac{1}{4}\cos 2x+C\)
36. Diketahui dari \(\int_{1}^{P}\limits \left ( x-1\right )^{2}dx = 2\frac{2}{3}\) . Nilai \(P\) yang memenuhi adalah
a) \(1\)
b) \(1\frac{1}{3}\)
c) \(3\)
d) \(6\)
e) \(9\)
Pembahasan:
\(\int_{1}^{p}u^{2}du = 2\frac{2}{3}\left [ \frac{1}{3}u^{3} \right ]_{1}^{p}=\frac{8}{3}\)
\(\left [\frac{1}{3}\left ( x-1 \right )^{3} \right ]_{1}^{P}=\frac{8}{3}\)
\(\frac{1}{3}\left ( p-1 \right )^{3}=\frac{8}{3}\)
\(\left ( p-1 \right )^{3}=\frac{8}{3}\times 3\)
\(\left ( p-1 \right )^{3}=8\)
\(\left ( p-1 \right )^{3}=2^{3}\)
\(\left ( p-1 \right )=2\)
\(p=3\)
37. Hasil dari \(\int\limits \cos ^{2}x\sin x dx\) adalah
a) \(\frac{1}{3}\cos ^{3}x+C\)
b) \(-\frac{1}{3}\cos ^{3}x+C\)
c) \(-\frac{1}{3}\sin ^{3}x+C\)
d) \(\frac{1}{3}\sin ^{3}x+C\)
e) \(3\sin ^{3}x+C\)
Pembahasan:
\(\int \cos ^{2}x\sin x dx\) misal \(A=\cos x \frac{dA}{dx}=-\sin x\Leftrightarrow dA=-\sin x dx\int \cos ^{2}x\sin x dx -\int A^{2}dA-\frac{1}{3}A^{3}+C-\frac{1}{3}\cos ^{3}x+C\)
38. Luas daerah yang dibatasi oleh kurva \(y=-x^{2}+4x\), sumbu \(X\), garis \(x=1\), dan \(x=3\) adalah
a) \(3\frac{2}{3} \)satuan luas
b) \(5\frac{1}{3}\) satuan luas
c) \(7\frac{1}{3}\) satuan luas
d) \(9\frac{1}{3}\) satuan luas
e) \(10\frac{2}{3}\) satuan luas
Pembahasan:
\(L=\int_{1}^{3}\left ( -x^{2}+4x \right )dx = \left [-\frac{1}{3}x^{3}+2x^{2} \right ]_{1}^{3}=\left ( -\frac{1}{3}\times 27+2\times 9 \right )-\left ( -\frac{1}{3}+2 \right )=\left ( -9+18 \right )-\left ( -\frac{1}{3}+2 \right )= 9-\frac{5}{3}=7\frac{1}{3}\)
39. hasil \(\int_{1}^{4}\limits \frac{2}{x\sqrt{x}}dx =\)
a) \(-12\)
b) \(-4\)
c) \(-3\)
d) \(2\)
e) \(\frac{3}{2}\)
Pembahasan:
\(\int_{1}^{4}2x^{-\frac{3}{2}}dx= \left [-4x^{-\frac{1}{2}} \right ]_{1}^{4}= \left [-\frac{4}{\sqrt{x}} \right ]_{1}^{4}=-\frac{4}{2}-\left ( -\frac{4}{1} \right )=-2+4=2\)
40. Volume benda putar yang terbentuk jika daerah yang dibatasi oleh kurva \(x-y^{2}+1=0\), \(x=1\), dan \(x=3\) adalah
a) \(8\frac{1}{2}\pi\) satuan volum
b) \(9\frac{1}{2}\pi\) satuan volum
c) \(11\frac{1}{2}\pi\) satuan volum
d) \(12\frac{1}{2}\pi\) satuan volum
e) \(13\frac{1}{2}\pi\) satuan volum
Pembahasan:
\(V=\pi \int y^{2}dx=\pi \int_{-1}^{4}\left ( x+1 \right )dx = \pi \left [\left ( \frac{1}{2}x^{2}+x \right ) \right ]_{-1}^{4}= \pi \left \{ \left ( \frac{1}{2}4^{2}+4 \right )-\left ( \frac{1}{2}\left ( -1 \right )^{2}+\left ( -1 \right ) \right ) \right \}=\pi \left ( 12+\frac{1}{2} \right )=12\frac{1}{2}\pi\)
