Pertanyaan

∫ x 2 x − 1 d x = ...

$\int x 2 x - 1 d x =$ ...

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jakarta

Jawaban terverifikasi

Pembahasan

Kita cari hasil integral di atas dengan menggunakan integral parsial. Misalkan u = x dan dan Sehingga

Kita cari hasil integral di atas dengan menggunakan integral parsial.

Misalkan u = x dan $begin mathsize 14px style fraction numerator d v over denominator d x end fraction equals square root of 2 x minus 1 end root end style$

$begin mathsize 14px style fraction numerator d u over denominator d x end fraction equals 1 end style$

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row v equals cell integral fraction numerator d v over denominator d x end fraction d x end cell row blank equals cell integral square root of 2 x minus 1 end root d x end cell row blank equals cell integral open parentheses 2 x minus 1 close parentheses to the power of 1 half end exponent d x end cell row blank equals cell 2 over 3 times open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent times 1 half plus straight C end cell row blank equals cell 1 third open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent plus C end cell end table end style$

Sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral x square root of 2 x minus 1 end root d x end cell equals cell integral u times fraction numerator d v over denominator d x end fraction d x end cell row blank equals cell u times v minus integral v times fraction numerator d u over denominator d x end fraction d x end cell row blank equals cell x times 1 third open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent minus integral 1 third open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent d x end cell row blank equals cell 1 third x open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent minus 1 over 15 open parentheses 2 x minus 1 close parentheses to the power of 5 over 2 end exponent plus C end cell row blank equals cell 1 third open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent open parentheses x minus 1 fifth open parentheses 2 x minus 1 close parentheses close parentheses plus C end cell row blank equals cell 1 third open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent open parentheses x minus 2 over 5 x plus 1 fifth close parentheses plus C end cell row blank equals cell 1 third open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent open parentheses 3 over 5 x plus 1 fifth close parentheses plus C end cell row blank equals cell 1 over 15 open parentheses 2 x minus 1 close parentheses to the power of 3 over 2 end exponent open parentheses 3 x plus 1 close parentheses plus C end cell end table end style$