Pertanyaan

∫ x + 1 x d x = ...

$\int \frac{x}{x + 1} d x = ...$

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Jawaban terverifikasi

Jawaban

hasilnya adalah .

hasilnya adalah $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank integral end table fraction numerator x over denominator x plus 1 end fraction d x table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank ln end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell vertical line x plus 1 vertical line end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table end style$ .

Pembahasan

Untuk menyelesaikan integral tersebut, dapat menggunakan metode substitusi. Misalkan, Kemudian substitusikan, sehingga didapat: Substitusikan kembali . Jadi, hasilnya adalah .

Untuk menyelesaikan integral tersebut, dapat menggunakan metode substitusi.

Misalkan,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell x plus 1 end cell row cell fraction numerator d u over denominator d x end fraction end cell equals 1 row cell d x end cell equals cell d u end cell row blank blank blank row u equals cell x plus 1 rightwards arrow x equals u minus 1 end cell end table end style$

Kemudian substitusikan, sehingga didapat:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator x over denominator x plus 1 end fraction d x end cell equals cell integral fraction numerator u minus 1 over denominator u end fraction d u end cell row blank equals cell integral 1 minus 1 over u d u end cell row blank equals cell u minus ln open vertical bar u close vertical bar plus C end cell end table end style$

Substitusikan kembali .

Jadi, hasilnya adalah $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank integral end table fraction numerator x over denominator x plus 1 end fraction d x table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank ln end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell vertical line x plus 1 vertical line end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table end style$ .