Contoh Soal Integral Fungsi Aljabar beserta Pembahasannya #4

 

Hai sob, pada postingan kali ini, mimin sajikan lanjutan beberapa contoh soal dan pembahasan pada materi integral fungsi aljabar (kelas 11 SMA). Cuss, langsung saja. Berikut contoh-contoh soal dan pembahasannya. Selamat belajar. Semoga bermanfaat.

Contoh soal 1 

Nilai dari ${\displaystyle \int x^2\sqrt{2x^3-1}dx}$ adalah ...

Jawab:

$\begin{array}{rl}{\displaystyle \int x^2\sqrt{2x^3-1}dx}& =\int x^2{(2x^3-1)}^{\frac{1}{2}}dx\end{array}$

Misalkan $u=2x^3-1$, maka $du=6x^{2}dx\iff dx=\frac{du}{6x^2}$

Sehingga, 

$\begin{array}{rl}{\displaystyle \int x^2{(2x^3-1)}^{\frac{1}{2}}dx}& =\int x^2{u}^{\frac{1}{2}}\frac{du}{6x^2}\\ & =\frac{1}{6}\int u^{\frac{1}{2}}du\\ & =\frac{1}{6}.\frac{2}{3}{u}^{\frac{3}{2}}+C\\ & =\frac{1}{9}{u}^{\frac{3}{2}}+C\\ & =\frac{1}{9}(2x^3-1{)}^{\frac{3}{2}}+C\end{array}$

Contoh soal 2 

Nilai dari ${\displaystyle \int \frac{x+1}{\sqrt{x^2+2x-2}}dx}$ adalah ...

Jawab:

$\begin{array}{rl}{\displaystyle \int \frac{x+1}{\sqrt{x^2+2x-2}}dx}& =\int \frac{x+1}{{(x^2+2x-2)}^{\frac{1}{2}}}dx\end{array}$

Misalkan $u=x^2+2x-2$, maka $du=2x+2dx\iff dx=\frac{du}{2x+2}$

Sehingga, 

$\begin{array}{rl}{\displaystyle \int \frac{x+1}{{(x^2+2x-2)}^{\frac{1}{2}}}dx}& =\int \frac{x+1}{u^{\frac{1}{2}}}\frac{du}{2x+2}\\ & =\int \frac{x+1}{u^{\frac{1}{2}}}\frac{du}{2(x+1)}\\ & =\frac{1}{2}\int \frac{1}{u^{\frac{1}{2}}}du\\ & =\frac{1}{2}\int u^{-\frac{1}{2}}du\\ & =\frac{1}{2}.2u^{\frac{1}{2}}+C\\ & =u^{\frac{1}{2}}+C\\ & ={(x^2+2x-2)}^{\frac{1}{2}}+C\end{array}$

Contoh soal 3

Hasil substitusi $u=x+1$ pada ${\displaystyle {\int }_0^1{x}^2\sqrt{x+1}dx}$ adalah ...

Jawab:

$\begin{array}{rl}{\displaystyle {\int }_0^1{x}^2\sqrt{x+1}dx}& ={\displaystyle {\int }_0^1{x}^2{(x+1)}^{\frac{1}{2}}dx}\end{array}$

Karena $u=x+1\iff x=u-1$, maka $du=dx$

Sehingga, 

$\begin{array}{rl}{\displaystyle {\int }_0^1{x}^2{(x+1)}^{\frac{1}{2}}dx}& ={\int }_0^1{x}^2{(u)}^{\frac{1}{2}}dx\\ & ={\int }_0^1(u-1{)}^2{(u)}^{\frac{1}{2}}du\end{array}$

Contoh soal 4

Nilai dari ${\displaystyle \int \frac{x}{\sqrt{9-x^2}}dx}$ adalah ...

Jawab:

$\begin{array}{rl}{\displaystyle \int \frac{x}{\sqrt{9-x^2}}dx}& =\int \frac{x}{(9-x^2{)}^{\frac{1}{2}}}dx\end{array}$

Misalkan $u=9-x^2$, maka $du=-2xdx\iff dx=-\frac{du}{2x}$

Sehingga, 

$\begin{array}{rl}\int \frac{x}{(9-x^2{)}^{\frac{1}{2}}}dx& =-\int \frac{x}{u^{\frac{1}{2}}}\frac{du}{2x}\\ & =-\frac{1}{2}\int u^{-\frac{1}{2}}du\\ & =-\frac{1}{2}.2u^{\frac{1}{2}}+C\\ & =-u^{\frac{1}{2}}+C\\ & =-(9-x^2{)}^{\frac{1}{2}}+C\end{array}$

Contoh soal 5

Nilai dari ${\displaystyle {\int }_0^3\frac{x}{\sqrt{25-x^2}}dx}$ adalah ...

Jawab:

$\begin{array}{rl}{\displaystyle {\int }_0^3\frac{x}{\sqrt{25-x^2}}dx}& ={\int }_0^3\frac{x}{{(25-x^2)}^{\frac{1}{2}}}dx\end{array}$

Misalkan $u=25-x^2$, maka $du=-2xdx\iff dx=-\frac{du}{2x}$

Sehingga, 

$\begin{array}{rl}{\displaystyle {\int }_0^3\frac{x}{{(25-x^2)}^{\frac{1}{2}}}dx}& =-{\int }_0^3\frac{x}{u^{\frac{1}{2}}}\frac{du}{2x}\\ & =-\frac{1}{2}{\int }_0^3\frac{1}{u^{\frac{1}{2}}}du\\ & =-(25-x^2{)}^{\frac{1}{2}}{|}_0^3\\ & =[-(25-3^2{)}^{\frac{1}{2}}]-[-(25-0^2{)}^{\frac{1}{2}}]\\ & =-4+5\\ & =1\end{array}$

Demikianlah beberapa contoh soal dan pembahasan pada materi integral fungsi aljabar. Semoga bermanfaat. 

Referensi:

Tim BBM. 2015. Big Book Matematika SMA Kelas 1, 2, dan 3. Jakarta: Cmedia.