Pertanyaan

Selesaikan soal berikut dengan substitusi! ∫ ( x 2 + 4 x − 5 ) 3 2 x + 4 d x

Selesaikan soal berikut dengan substitusi!

$\int \frac{2 x + 4}{( x ^{2} + 4 x - 5 ) ^{3}} d x$

F. Kurnia

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

.

$begin mathsize 14px style integral fraction numerator 2 x plus 4 over denominator open parentheses x squared plus 4 x minus 5 close parentheses cubed end fraction d x end style$ $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 over denominator 2 left parenthesis x squared plus 4 x minus 5 right parenthesis squared end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table end style$ .

Pembahasan

Misalkan: , maka bentuk aljabar pada integral dapat dituliskan sebagai berikut Substitusikan pemisalan u , pada hasil integral Jadi, .

Misalkan: , maka bentuk aljabar pada integral dapat dituliskan sebagai berikut

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral 1 over u cubed d u end cell equals cell integral u to the power of negative 3 end exponent d u end cell row blank equals cell fraction numerator 1 over denominator negative 2 u apostrophe end fraction u to the power of negative 2 end exponent plus C end cell row blank blank blank row blank blank blank end table end style$

Substitusikan pemisalan u, pada hasil integral

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative fraction numerator u to the power of negative 2 end exponent over denominator negative 2 end fraction plus C end cell equals cell fraction numerator left parenthesis x squared plus 4 x minus 5 right parenthesis to the power of negative 2 end exponent over denominator negative 2 end fraction plus C end cell row blank equals cell negative fraction numerator 1 over denominator 2 left parenthesis x squared plus 4 x minus 5 right parenthesis squared end fraction plus C end cell end table end style$

Jadi, $begin mathsize 14px style integral fraction numerator 2 x plus 4 over denominator open parentheses x squared plus 4 x minus 5 close parentheses cubed end fraction d x end style$ $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 over denominator 2 left parenthesis x squared plus 4 x minus 5 right parenthesis squared end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table end style$ .