Lanjutan Kumpulan Contoh Soal Integral (1)

\begin{array}{ll}\\ \fbox{11}.&\displaystyle \int x^{2}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle \frac{1}{3}x^{3}+C\\ &b.\quad \displaystyle \frac{1}{4}x^{6}+C\\ &c.\quad \displaystyle \frac{1}{3}x^{6}+C\\ &d.\quad \displaystyle \frac{1}{6}x^{3}+C\\ &e.\quad \displaystyle \frac{2}{3}x^{3}+C\end{array}.

Jawab:

\displaystyle \int x^{2}\: dx=\displaystyle \frac{x^{2+1}}{2+1}+C= \displaystyle \frac{1}{3}x^{3}+C.

\begin{array}{ll}\\ \fbox{12}.&\displaystyle \int \frac{1}{3}x^{3}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle \frac{1}{3}x^{4}+C\\ &b.\quad \displaystyle \frac{1}{4}x^{4}+C\\ &c.\quad \displaystyle x^{4}+C\\ &d.\quad \displaystyle \frac{1}{12}x^{4}+C\\ &e.\quad \displaystyle \frac{4}{3}x^{4}+C\end{array}.

Jawab:

\displaystyle \int \frac{1}{3}x^{3}\: dx=\displaystyle \frac{1}{3}.\frac{x^{3+1}}{3+1}+C= \displaystyle \frac{1}{12}x^{4}+C.

\begin{array}{ll}\\ \fbox{13}.&\displaystyle \int x^{-2}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle -2x^{-1}+C\\ &b.\quad \displaystyle -x^{-1}+C\\ &c.\quad \displaystyle -\frac{1}{2}x^{-2}+C\\ &d.\quad \displaystyle -\frac{1}{3}x^{-3}+C\\ &e.\quad \displaystyle -3x^{-3}+C\end{array}.

Jawab:

\displaystyle \int x^{-2}\: dx=\displaystyle \frac{x^{-2+1}}{-2+1}+C= \displaystyle \frac{1}{-1}x^{-1}+C=-x^{-1}+C.

\begin{array}{ll}\\ \fbox{14}.&\displaystyle \int x^{\frac{1}{3}}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle \frac{3}{4}x^{\frac{4}{3}}+C\\ &b.\quad \displaystyle x^{\frac{4}{3}}+C\\ &c.\quad \displaystyle \frac{3}{4}x^{\frac{2}{3}}+C\\ &d.\quad \displaystyle x^{-\frac{2}{3}}+C\\ &e.\quad \displaystyle \frac{3}{4}x^{-\frac{2}{3}}+C\end{array}.

Jawab:

\displaystyle \int x^\frac{1}{3}\: dx=\displaystyle \frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}+C= \displaystyle \frac{1}{\frac{4}{3}}x^{\frac{4}{3}}+C=\displaystyle \frac{3}{4}x^{\frac{4}{3}}+C.

\begin{array}{ll}\\ \fbox{15}.&\displaystyle \int \frac{1}{x^{3}}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle -\frac{1}{2x^{2}}+C\\ &b.\quad \displaystyle -\frac{2}{x^{2}}+C\\ &c.\quad \displaystyle \frac{1}{3x^{4}}+C\\ &d.\quad \displaystyle \frac{3}{x^{4}}+C\\ &e.\quad \displaystyle -\frac{1}{4x^{3}}+C\end{array}.

Jawab:

\displaystyle \int \frac{1}{x^{3}}\: dx=\int x^{-3}\: dx=\displaystyle \frac{x^{-3+1}}{-3+1}+C=\frac{x^{-2}}{-2}+C=-\frac{1}{2x^{2}}+C.

\begin{array}{ll}\\ \fbox{16}.&\displaystyle \int x^{3}\sqrt{x}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle \frac{2}{9}x^{4}\sqrt{x}+C\\ &b.\quad \displaystyle \frac{9}{2}x^{4}\sqrt{x}+C\\ &c.\quad \displaystyle \frac{1}{9}x^{4}\sqrt{x}+C\\ &d.\quad \displaystyle 9x^{4}\sqrt{x}+C\\ &e.\quad \displaystyle x^{4}\sqrt{x}+C\end{array}.

Jawab:

\begin{aligned}\displaystyle \int x^{3}\sqrt{x}\: dx=\int x^{3}.x^{\frac{1}{2}}\: dx=\int x^{3\frac{1}{2}}\: dx=\int x^\frac{7}{2}\: dx=\displaystyle \frac{x^{\frac{7}{2}+1}}{\frac{7}{2}+1}+C=\displaystyle \frac{x^{\frac{9}{2}}}{\frac{9}{2}}+C=\displaystyle \frac{2}{9}x^{4\frac{1}{2}}+C=\displaystyle \frac{2}{9}x^{4}\sqrt{x}+C \end{aligned}.

\begin{array}{ll}\\ \fbox{17}.&\displaystyle \int x\sqrt{x\sqrt[3]{x^{2}}}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle \frac{17}{6}x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C\\ &b.\quad \displaystyle \frac{6}{17}x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C\\ &c.\quad \displaystyle x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C\\ &d.\quad \displaystyle \frac{6}{17}x\sqrt{x\sqrt[3]{x^{2}}}+C\\ &e.\quad \displaystyle \frac{1}{2}x\sqrt{x\sqrt[3]{x^{2}}}+C\end{array}.

Jawab:

\begin{aligned}\displaystyle \int x\sqrt{x\sqrt[3]{x^{2}}}\: dx&=\int x\left ( x.x^{\frac{2}{3}} \right )^{\frac{1}{2}}\: dx=\int x^{1+\frac{1}{2}+\frac{2}{6}}\: dx=\int x^{\frac{11}{6}}\: dx=\displaystyle \frac{x^{\frac{11}{6}+1}}{\frac{11}{6}+1}+C=\displaystyle \frac{x^{\frac{17}{6}}}{\frac{17}{6}}+C=\displaystyle \frac{6}{17}x^{2}.x^{\frac{5}{6}}+C\\ &=\displaystyle \frac{6}{17}x^{2}\left ( x.x^{\frac{2}{3}} \right )^{\frac{1}{2}}+C\\ &=\displaystyle \frac{6}{17}x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C \end{aligned}.

\begin{array}{ll}\\ \fbox{18}.&\displaystyle \int x^{2}\sqrt{x^{2}\sqrt[3]{x^{3}}}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ &b.\quad \displaystyle \frac{27}{6}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ &c.\quad \displaystyle \frac{6}{27}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ &d.\quad \displaystyle \frac{6}{21}x^{2}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ &e.\quad \displaystyle \frac{6}{21}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\end{array}.

Jawab:

\begin{aligned}\displaystyle \int x^{2}\sqrt{x^{2}\sqrt[3]{x^{3}}}\: dx&=\int x^{2}\left ( x^{2}.x^{1} \right )^{\frac{1}{2}}\: dx=\int x^{2+\frac{2}{2}+\frac{1}{2}}\: dx=\int x^{\frac{7}{2}}\: dx=\displaystyle \frac{x^{\frac{7}{2}+1}}{\frac{7}{2}+1}+C=\displaystyle \frac{x^{\frac{9}{2}}}{\frac{9}{2}}+C\\ &=\displaystyle \frac{2}{9}x^{\frac{9}{2}}+C\\ &=\displaystyle \frac{2.3}{9.3}x^{3}.x^{\frac{3}{2}}+C\\ &=\displaystyle \frac{6}{27}x^{3}\left ( x^{3} \right )^{\frac{1}{2}}+C\\ &=\displaystyle \frac{6}{27}x^{3}\sqrt{x^{2}.x}+C\\ &=\displaystyle \frac{6}{27}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\end{aligned}.

\begin{array}{ll}\\ \fbox{19}.&\displaystyle \int x^{3}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle \frac{1}{4}x^{4}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ &b.\quad \displaystyle \frac{6}{23}x^{4}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ &c.\quad \displaystyle \frac{23}{6}x^{4}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ &d.\quad \displaystyle \frac{23}{6}x^{3}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ &e.\quad \displaystyle \frac{3}{4}x^{3}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\end{array}.

Jawab:

\begin{aligned}\displaystyle \int x^{3}\sqrt{\frac{1}{x}\sqrt[3]{x^{2}}}\: dx&=\int x^{3}\left ( x^{-1}.x^{\frac{2}{3}} \right )^{\frac{1}{2}}\: dx=\int x^{3-\frac{1}{2}+\frac{1}{3}}\: dx=\int x^{\frac{17}{6}}\: dx=\displaystyle \frac{x^{\frac{17}{6}+1}}{\frac{17}{6}+1}+C=\displaystyle \frac{x^{\frac{23}{6}}}{\frac{23}{6}}+C\\ &=\displaystyle \frac{6}{23}x^{\frac{24-1}{6}}+C\\ &=\displaystyle \frac{6}{23}x^{4}.x^{-\frac{1}{6}}+C\\ &=\displaystyle \frac{6}{23}x^{4}.\left ( x^{-\frac{1}{3}} \right )^{\frac{1}{2}}+C\\ &=\displaystyle \frac{6}{23}x^{4}\sqrt{x^{-1+\frac{2}{3}}}+C\\ &=\displaystyle \frac{6}{23}x^{4}\sqrt{\frac{1}{x}\sqrt[3]{x^{2}}}+C\end{aligned}.

\begin{array}{ll}\\ \fbox{20}.&\displaystyle \int x\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}\: dx=\end{array}\\ \begin{array}{ll}\\ .\: \: \qquad &a.\quad \displaystyle x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ &b.\quad \displaystyle \frac{13}{8}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ &c.\quad \displaystyle \frac{8}{13}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ &d.\quad \displaystyle \frac{1}{2}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ &e.\quad \displaystyle \frac{8}{5}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\end{array}.

Jawab:

\begin{aligned}\displaystyle \int x\sqrt{\frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}\: \: dx&=\int x\left ( x^{-1}\left ( x\left ( x^{-1} \right )^{\frac{1}{2}} \right )^{\frac{1}{2}} \right )^{\frac{1}{2}} \: dx=\int x\left ( x^{-\frac{1}{2}}.x^{\frac{1}{4}}.x^{-\frac{1}{8}} \right )\: dx=\int x^{1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}}\: dx=\int x^{\frac{5}{8}}\: dx\\ &=\displaystyle \frac{x^{\frac{5}{8}+1}}{\frac{5}{8}+1}+C\\ &=\displaystyle \frac{x^{\frac{13}{8}}}{\frac{13}{8}}+C\\ &=\displaystyle \frac{8}{13}x^{\frac{16-3}{8}}+C\\ &=\displaystyle \frac{8}{13}x^{2}.x^{-\frac{3}{8}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{x^{-\frac{3}{4}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{x^{-1+\frac{1}{4}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{\frac{1}{x}.\left ( x^{\frac{1}{2}} \right )^{\frac{1}{2}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{\frac{1}{x}\sqrt{x^{1-\frac{1}{2}}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{\frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\end{aligned}.

Sumber Referensi

  1. Kuntarti, Sulistiyono dan Sri Kurnianingsih. 2007. Matematika  SMA dan MA untuk Kelas XII Semester 1 Program IPA Standar Isi 2006. Jakarta: esis.
  2. Susianto, Bambang. 2011. Olimpiade Matematika dengan Proses Berpikir Aljabar dan Bilangan. Jakarta: Grasindo.
  3. Setiawan, Tedy, Margana, Edi Kusnaedi. 2007. Seri Integral 1000 Bank Soal SMA/MA. Bandung: Yrama Widya.
  4. Tung, Khoe Yao. 2012. Pintar Matematika SMA Kelas XII IPA untuk Olimpiade dan Pengayaan Pelajaran. Yogyakarta: ANDI.

Tentang ahmadthohir1089

Nama saya Ahmad Thohir, asli orang Purwodadi lahir di Grobogan 02 Februari 1980. Pendidikan : Tingkat dasar lulus dari MI Nahdlatut Thullab di desa Manggarwetan,kecamatan Godong lulus tahun 1993. dan untuk tingkat menengah saya tempuh di MTs Nahdlatut Thullab Manggar Wetan lulus tahun 1996. Sedang untuk tingkat SMA saya menamatkannya di MA Futuhiyyah-2 Mranggen, Demak lulus tahun 1999. Setelah itu saya Kuliah di IKIP PGRI Semarang pada fakultas FPMIPA Pendidikan Matematika lulus tahun 2004. Pekerjaan : Sebagai guru (PNS DPK Kemenag) mapel matematika di MA Futuhiyah Jeketro, Gubug. Pengalaman mengajar : 1. GTT di MTs Miftahul Mubtadiin Tambakan Gubug tahun 2003 s/d 2005 2. GTT di SMK Negeri 3 Semarang 2005 s/d 2009 3. GT di MA Futuhiyah Jeketro Gubug sejak 1 September 2009
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