Pertanyaan

∫ 0 2 π cos x + 1 sin 2 x d x = ...

$\int_{0}^{\frac{π}{2}} \frac{sin 2 x}{cos x + 1} d x = ...$

S. Yoga

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

.

$integral subscript 0 superscript pi over 2 end superscript fraction numerator sin open parentheses 2 x close parentheses over denominator square root of cos open parentheses x close parentheses plus 1 end root end fraction d x equals 4 times fraction numerator 2 minus square root of 2 over denominator 3 end fraction$ .

Pembahasan

Ingat bahwa: Sehingga Misalkan: Sehingga Misalkan: Sehingga Jadi, .

Ingat bahwa:

Sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 0 superscript pi over 2 end superscript fraction numerator sin open parentheses 2 x close parentheses over denominator square root of cos open parentheses x close parentheses plus 1 end root end fraction straight d x end cell equals cell integral subscript 0 superscript pi over 2 end superscript fraction numerator 2 cos open parentheses x close parentheses sin open parentheses x close parentheses over denominator square root of 1 plus cos open parentheses x close parentheses end root end fraction straight d x end cell row blank equals cell 2 times integral subscript 0 superscript pi over 2 end superscript fraction numerator cos open parentheses x close parentheses sin open parentheses x close parentheses over denominator square root of 1 plus cos open parentheses x close parentheses end root end fraction straight d x end cell end table$

Misalkan:

$table attributes columnalign right center left columnspacing 0px end attributes row u equals cell cos open parentheses x close parentheses end cell row cell fraction numerator straight d u over denominator straight d x end fraction end cell equals cell negative sin space x end cell row cell straight d x end cell equals cell open parentheses negative fraction numerator 1 over denominator sin open parentheses x close parentheses end fraction close parentheses straight d u end cell end table$

Sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell 2 times integral subscript 1 superscript 0 minus fraction numerator u over denominator square root of 1 plus u end root end fraction straight d u end cell equals cell 2 open parentheses negative integral subscript 0 superscript 1 minus fraction numerator u over denominator square root of 1 plus u end root end fraction straight d u close parentheses end cell row blank equals cell 2 open parentheses negative open parentheses negative integral subscript 0 superscript 1 fraction numerator u over denominator square root of 1 plus u end root end fraction straight d u close parentheses close parentheses end cell end table$

Misalkan:

$table attributes columnalign right center left columnspacing 0px end attributes row v equals cell square root of 1 plus u end root end cell row cell d v end cell equals cell fraction numerator 1 over denominator 2 square root of 1 plus u end root end fraction d u end cell row cell d u end cell equals cell 2 square root of 1 plus u end root d v end cell end table$

Sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 1 superscript square root of 2 end superscript 2 open parentheses v squared minus 1 close parentheses straight d v end cell equals cell 2 open parentheses negative open parentheses negative integral subscript 1 superscript square root of 2 end superscript 2 open parentheses v squared minus 1 close parentheses straight d v close parentheses close parentheses end cell row blank equals cell 2 open parentheses negative open parentheses negative 2 times integral subscript 1 superscript square root of 2 end superscript v squared minus 1 straight d v close parentheses close parentheses end cell row blank equals cell 2 open parentheses negative open parentheses negative 2 open parentheses integral subscript 1 superscript square root of 2 end superscript v squared straight d v minus integral subscript 1 superscript square root of 2 end superscript 1 straight d v close parentheses close parentheses close parentheses end cell row blank equals cell 2 open parentheses negative open parentheses negative 2 open parentheses open square brackets fraction numerator v to the power of 2 plus 1 end exponent over denominator 2 plus 1 end fraction close square brackets subscript 1 superscript square root of 2 end superscript minus open square brackets 1 times thin space v close square brackets subscript 1 superscript square root of 2 end superscript close parentheses close parentheses close parentheses end cell row blank equals cell 2 open parentheses negative open parentheses negative 2 open parentheses open square brackets v cubed over 3 close square brackets subscript 1 superscript square root of 2 end superscript minus open square brackets v close square brackets subscript 1 superscript square root of 2 end superscript close parentheses close parentheses close parentheses end cell row blank equals cell 2 open parentheses negative open parentheses negative 2 open parentheses open parentheses fraction numerator 2 square root of 2 over denominator 3 end fraction minus 1 third close parentheses minus open parentheses square root of 2 minus 1 close parentheses close parentheses close parentheses close parentheses end cell row blank equals cell 4 open parentheses fraction numerator 2 square root of 2 over denominator 3 end fraction minus 1 third minus open parentheses square root of 2 minus 1 close parentheses close parentheses end cell row blank equals cell 4 times fraction numerator 2 minus square root of 2 over denominator 3 end fraction end cell end table$

Jadi, $integral subscript 0 superscript pi over 2 end superscript fraction numerator sin open parentheses 2 x close parentheses over denominator square root of cos open parentheses x close parentheses plus 1 end root end fraction d x equals 4 times fraction numerator 2 minus square root of 2 over denominator 3 end fraction$ .