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∫−2π​2π​​(sin2 x−2)⋅cos x dx= ...

Pertanyaan

integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript open parentheses sin squared space x minus 2 close parentheses times cos space x space d x equals space... 

S. Yoga

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

 integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript open parentheses sin squared open parentheses x close parentheses minus 2 close parentheses cos open parentheses x close parentheses straight d x equals negative 10 over 3.

Pembahasan

Ingat bahwa:

  • table attributes columnalign right center left columnspacing 0px end attributes row cell integral f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses d x end cell equals cell integral f open parentheses x close parentheses d x plus-or-minus integral g open parentheses x close parentheses d x end cell row cell integral x to the power of a straight d x end cell equals cell fraction numerator x to the power of a plus 1 end exponent over denominator a plus 1 end fraction comma space a not equal to negative 1 end cell row cell integral a f left parenthesis x right parenthesis straight d x end cell equals cell a integral f left parenthesis x right parenthesis straight d x end cell row cell integral cos space left parenthesis x right parenthesis straight d x end cell equals cell sin space x end cell end table 

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript open parentheses sin squared space x minus 2 close parentheses times cos space x space d x end cell equals cell integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript cos open parentheses x close parentheses sin squared open parentheses x close parentheses minus cos open parentheses x close parentheses times 2 space d x end cell row blank equals cell integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript open parentheses cos open parentheses x close parentheses sin squared open parentheses x close parentheses minus 2 cos open parentheses x close parentheses close parentheses space d x end cell row blank equals cell integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript cos open parentheses x close parentheses sin squared open parentheses x close parentheses d x minus integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript 2 cos open parentheses x close parentheses d x end cell end table 

Misalkan:

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell sin open parentheses x close parentheses end cell row cell straight d u end cell equals cell cos open parentheses x close parentheses straight d x end cell row cell straight d x end cell equals cell fraction numerator 1 over denominator cos open parentheses x close parentheses end fraction straight d u end cell row blank blank blank row u equals cell sin open parentheses x close parentheses end cell row x equals cell negative pi over 2 rightwards arrow u equals negative 1 end cell row x equals cell pi over 2 rightwards arrow u equals 1 end cell end table 

Sehingga,

integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript cos open parentheses x close parentheses sin squared open parentheses x close parentheses straight d x minus integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript 2 cos open parentheses x close parentheses straight d x equals integral subscript negative 1 end subscript superscript 1 u squared straight d u minus 2 times integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript cos open parentheses x close parentheses straight d x equals open square brackets u cubed over 3 close square brackets subscript negative 1 end subscript superscript 1 minus 2 open square brackets sin open parentheses x close parentheses close square brackets subscript negative pi over 2 end subscript superscript pi over 2 end superscript equals open square brackets 1 third minus open parentheses negative 1 third close parentheses close square brackets minus open square brackets 2 times thin space 2 close square brackets equals 2 over 3 minus 4 equals negative 10 over 3 

Jadi, integral subscript negative pi over 2 end subscript superscript pi over 2 end superscript open parentheses sin squared open parentheses x close parentheses minus 2 close parentheses cos open parentheses x close parentheses straight d x equals negative 10 over 3.

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