Hasil dari [tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx}[/tex] adalah [tex]\displaystyle{\boldsymbol{\frac{\sqrt{2}+ln|\sqrt{2}+1|}{2}} }[/tex].
PEMBAHASAN
Integral merupakan operasi yang menjadi kebalikan dari operasi turunan/diferensial. Sehingga integral sering juga disebut sebagai antiturunan.
[tex]\displaystyle{f(x)=\int\limits {\left [ \frac{df(x)}{dx} \right ]} \, dx}[/tex]
Salah satu metode untuk menyelesaikan integral adalah metode integral parsial. Dimana :
[tex]\displaystyle{\int\limits {u} \, dv=uv-\int\limits {v} \, du }[/tex]
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DIKETAHUI
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx= }[/tex]
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DITANYA
Tentukan hasilnya.
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PENYELESAIAN
Kita gunakan integral parsial.
[tex]\displaystyle{\int\limits {csc^3x} \, dx=\int\limits {csc^2x.cscx} \, dx }[/tex]
[tex]---------------[/tex]
Misal :
[tex]u=cscx~\to~du=-cscx.cotxdx[/tex]
[tex]dv=csc^2xdx~\to~v=-cotx[/tex]
[tex]---------------[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=uv-\int\limits {v} \, du }[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=-cscx.cotx-\int\limits {-cotx(-cscx.cotx)} \, dx }[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=-cscx.cotx-\int\limits {cot^2x.cscx} \, dx }[/tex]
[tex]---------------[/tex]
Gunakan identitas [tex]cot^2x+1=csc^2x[/tex]
[tex]---------------[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=-cscx.cotx-\int\limits {(csc^2x-1)cscx} \, dx }[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=-cscx.cotx-\int\limits {(csc^3x-cscx)} \, dx }[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=-cscx.cotx-\int\limits {csc^3x} \, dx+\int\limits {cscx} \, dx }[/tex]
[tex]\displaystyle{2\int\limits {csc^3x} \, dx=-cscx.cotx+\int\limits {cscx} \, dx }[/tex]
[tex]\displaystyle{2\int\limits {csc^3x} \, dx=-cscx.cotx-ln|cscx+cotx|+C }[/tex]
[tex]\displaystyle{\int\limits {csc^3x} \, dx=-\frac{1}{2}cscx.cotx-\frac{1}{2}ln|cscx+cotx|+C }[/tex]
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Maka :
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=-\frac{1}{2}cscx.cotx-\frac{1}{2}ln|cscx+cotx|\Bigr|^{\frac{\pi}{2}}_{\frac{\pi}{4}} }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=-\frac{1}{2}\frac{1}{sinx}\frac{cosx}{sinx}-\frac{1}{2}ln\left | \frac{1}{sinx}+\frac{cosx}{sinx} \right |\Bigr|^{\frac{\pi}{2}}_{\frac{\pi}{4}} }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=-\frac{cosx}{2sin^2x}-\frac{1}{2}ln\left | \frac{1+cosx}{sinx} \right |\Bigr|^{\frac{\pi}{2}}_{\frac{\pi}{4}} }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=\left [ -\frac{cos\frac{\pi}{2}}{2sin^2\frac{\pi}{2}}-\frac{1}{2}ln\left | \frac{1+cos\frac{\pi}{2}}{sin\frac{\pi}{2}} \right | \right ]-\left [ -\frac{cos\frac{\pi}{4}}{2sin^2\frac{\pi}{4}}-\frac{1}{2}ln\left | \frac{1+cos\frac{\pi}{4}}{sin\frac{\pi}{4}} \right | \right ] }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=\left [ -\frac{0}{2(1)^2}-\frac{1}{2}ln\left | \frac{1+0}{1} \right | \right ]-\left [ -\frac{\frac{\sqrt{2}}{2}}{2(\frac{\sqrt{2}}{2})^2}-\frac{1}{2}ln\left | \frac{1+\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \right | \right ] }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=\left [ 0-\frac{1}{2}ln(1) \right ]-\left [ -\frac{\sqrt{2}}{2}-\frac{1}{2}ln\left | \frac{2+\sqrt{2}}{\sqrt{2}} \right | \right ] }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=0+\frac{\sqrt{2}}{2}+\frac{1}{2}ln\left | \frac{2+\sqrt{2}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}} \right | }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=\frac{\sqrt{2}}{2}+\frac{1}{2}ln\left | \frac{2\sqrt{2}+2}{2} \right | }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=\frac{\sqrt{2}}{2}+\frac{1}{2}ln\left | \sqrt{2}+1 \right | }[/tex]
[tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx=\frac{\sqrt{2}+ln|\sqrt{2}+1|}{2} }[/tex]
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KESIMPULAN
Hasil dari [tex]\displaystyle{\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {csc^3x} \, dx}[/tex] adalah [tex]\displaystyle{\boldsymbol{\frac{\sqrt{2}+ln|\sqrt{2}+1|}{2}} }[/tex].
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PELAJARI LEBIH LANJUT
- Integral parsial : https://brainly.co.id/tugas/30673657
- Integral parsial : https://brainly.co.id/tugas/28945863
- Integral metode substitusi : https://brainly.co.id/tugas/30176534
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DETAIL JAWABAN
Kelas : 11
Mapel: Matematika
Bab : Integral Tak Tentu
Kode Kategorisasi: 11.2.10
Kata Kunci : integral, anti turunan, parsial.
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