Pertanyaan

∫ ( 3 x 4 − 2 x 2 + 1 ) 3 x ( 3 x 2 − 1 ) d x = ...

$\int \frac{x ( 3 x ^{2} - 1 )}{( 3 x ^{4} - 2 x ^{2} + 1 ) ^{3}} d x = ...$

$fraction numerator size 12px minus size 12px 2 over denominator square root of size 12px 3 size 12px x to the power of size 12px 4 size 12px minus size 12px 2 size 12px x to the power of size 12px 2 size 12px plus size 12px 1 end root end fraction size 12px plus size 12px C$
$fraction numerator size 12px minus size 12px 2 over denominator size 12px 3 square root of size 12px 3 size 12px x to the power of size 12px 4 size 12px minus size 12px 2 size 12px x to the power of size 12px 2 size 12px plus size 12px 1 end root end fraction size 12px plus size 12px C$
$fraction numerator size 12px minus size 12px 1 over denominator size 12px 2 square root of size 12px 3 size 12px x to the power of size 12px 4 size 12px minus size 12px 2 size 12px x to the power of size 12px 2 size 12px plus size 12px 1 end root end fraction size 12px plus size 12px C$

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Pembahasan

Gunakan integral substitusi Jadi,

Gunakan integral substitusi

$begin mathsize 12px style u equals 3 x to the power of 4 minus 2 x squared plus 1 fraction numerator d u over denominator d x end fraction equals 12 x cubed minus 4 x d x equals fraction numerator d u over denominator 4 x open parentheses 3 x squared minus 1 close parentheses end fraction end style$

Jadi,

$begin mathsize 12px style integral fraction numerator x open parentheses 3 x squared minus 1 close parentheses over denominator open parentheses 3 x to the power of 4 minus 2 x squared plus 1 close parentheses cubed end fraction d x equals integral fraction numerator x open parentheses 3 x squared minus 1 close parentheses over denominator square root of u cubed end root end fraction fraction numerator d u over denominator 4 x open parentheses 3 x squared minus 1 close parentheses end fraction equals integral 1 fourth u to the power of negative 3 over 2 end exponent d u equals 1 fourth times open parentheses negative 2 close parentheses u to the power of negative 1 half end exponent plus C equals negative 1 half times fraction numerator 1 over denominator square root of u end fraction plus C equals negative fraction numerator 1 over denominator 2 square root of 3 x to the power of 4 minus 2 x squared plus 1 end root end fraction plus C end style$

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