Hasil pengintegralan fungsi aljabar ∫12(6x+3)x2+x−2dx adalah ....

Pertanyaan

Hasil pengintegralan fungsi aljabar $\style{font-size:14px}{\int_1^2(6x+3)\sqrt{x^2+x-2}\operatorname dx}$ adalah .... $\;$

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... $undefined$

Pembahasan Soal:

Integral fungsi pada soal dapat dikerjakan dengan metode substitusi. Misalkan $begin mathsize 14px style x squared plus x minus 2 equals p end style$ maka dengan menurunkan kedua ruas diperoleh $begin mathsize 14px style left parenthesis 2 x plus 1 right parenthesis d x equals d p end style$ sehingga $begin mathsize 14px style d x equals fraction numerator 1 over denominator 2 x plus 1 end fraction d p end style$ .

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript blank left parenthesis 6 x plus 3 right parenthesis square root of x squared plus x minus 2 end root d x end cell equals cell integral subscript blank left parenthesis 6 x plus 3 right parenthesis square root of p times fraction numerator 1 over denominator 2 x plus 1 end fraction d p end cell row blank equals cell 3 integral subscript blank square root of p d p end cell row blank equals cell fraction numerator 3 over denominator begin display style 1 half end style plus 1 end fraction p to the power of 1 half plus 1 end exponent plus C end cell row blank equals cell 2 p square root of p plus C end cell end table end style$

dengan $begin mathsize 14px style C end style$ konstanta. Selanjutnya dengan mensubstitusi kembali nilai $begin mathsize 14px style p end style$ , diperoleh

sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 1 superscript 2 left parenthesis 6 x plus 3 right parenthesis square root of x squared plus x minus 2 end root d x end cell equals cell 2 left parenthesis x squared plus x minus 2 right parenthesis square root of x squared plus x minus 2 end root left enclose blank with 1 below and 2 on top end enclose end cell row blank equals cell open square brackets 2 left parenthesis 2 squared plus 2 minus 2 right parenthesis square root of 2 squared plus 2 minus 2 end root close square brackets end cell row blank blank cell space space space minus open square brackets 2 left parenthesis 1 squared plus 1 minus 2 right parenthesis square root of 1 squared plus 1 minus 2 end root close square brackets end cell row blank equals cell 16 minus 0 end cell row blank equals 16 end table end style$

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Rizky

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 05 Oktober 2021