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Jika , hitung nilai integralnya, selesaikan menggu...

Jika begin mathsize 14px style integral subscript 1 superscript 2 open parentheses x squared plus 7 x minus 5 close parentheses space cos space open parentheses 2 x close parentheses space d x end style, hitung nilai integralnya, selesaikan menggunakan integral parsial

Jawaban:

Kita cari hasil integral di atas dengan menggunakan metode integral parsial. Adapun rumus integral parsial adalah :

begin mathsize 14px style integral u space d v equals u space v minus integral v space d u end style 

pilih fungsi yang saat diturunkan sampai nol sebagai pemisalan begin mathsize 14px style u end style

Misal :

begin mathsize 14px style u equals x squared plus 7 x minus 5 end style

begin mathsize 14px style v equals integral cos space open parentheses 2 x close parentheses space d x end style 

Cari turunan dari begin mathsize 14px style u end style terlebih dahulu :

begin mathsize 14px style u apostrophe equals fraction numerator d u over denominator d x end fraction fraction numerator d over denominator d x end fraction open parentheses x squared plus 7 x minus 5 close parentheses equals 2 x plus 7 fraction numerator d u over denominator d x end fraction equals 2 x plus 7 d u equals 2 x plus 7 space d x end style  

Cari integral dari begin mathsize 14px style v end style terlebih dahulu :

begin mathsize 14px style v equals integral cos space open parentheses 2 x close parentheses space d x m i s a l space colon u equals 2 x fraction numerator d u over denominator d x end fraction equals 2 d x equals 1 half d u . integral cos space left parenthesis 2 x right parenthesis space d x equals 1 half integral cos space u space d u integral c o s space left parenthesis 2 x right parenthesis space d x equals 1 half space s in space u integral c o s space left parenthesis 2 x right parenthesis space d x equals 1 half space s in space open parentheses 2 x close parentheses  end style 

Substitusi pada rumus integral parsial :

begin mathsize 14px style integral u space d v equals u space v minus integral v space d u integral u space d v equals open parentheses x squared plus 7 x minus 5 close parentheses space open parentheses 1 half space sin space open parentheses 2 x close parentheses close parentheses minus integral open parentheses 1 half space sin space open parentheses 2 x close parentheses close parentheses open parentheses 2 x plus 7 close parentheses space d x end style  

Kita lakukan integral parsial kembali untuk :

begin mathsize 14px style integral open parentheses 2 x plus 7 close parentheses open parentheses sin open parentheses 2 x close parentheses close parentheses space d x end style 

misal :

begin mathsize 14px style u equals left parenthesis 2 x plus 7 right parenthesis fraction numerator d u over denominator d x end fraction equals 2 end style        begin mathsize 14px style d v equals sin open parentheses 2 x close parentheses space space v equals integral sin space open parentheses 2 x close parentheses space space v equals negative 1 half cos open parentheses 2 x close parentheses end style 

sehingga :

begin mathsize 14px style equals u space v minus integral v space d u equals open parentheses 2 x plus 7 close parentheses space open parentheses negative 1 half close parentheses open parentheses cos space open parentheses 2 x close parentheses close parentheses minus integral open parentheses negative 1 half close parentheses open parentheses cos open parentheses 2 x close parentheses close parentheses space 2 space d x equals open parentheses fraction numerator open parentheses negative 2 x minus 7 close parentheses over denominator 2 end fraction close parentheses open parentheses cos space open parentheses 2 x close parentheses close parentheses plus integral open parentheses cos open parentheses 2 x close parentheses close parentheses space d x equals open parentheses fraction numerator open parentheses negative 2 x minus 7 close parentheses over denominator 2 end fraction close parentheses open parentheses cos space open parentheses 2 x close parentheses close parentheses plus 1 half sin open parentheses 2 x close parentheses end style 

Kemudian substitusikan hasil integral parsial ke-2 ke integrial parsial pertama.

 begin mathsize 14px style integral u space d v equals u space v minus integral v space d u integral u space d v equals left parenthesis x squared plus 7 x minus 5 right parenthesis space left parenthesis 1 half space sin space open parentheses 2 x close parentheses right parenthesis minus integral left parenthesis 1 half space sin space open parentheses 2 x close parentheses right parenthesis left parenthesis 2 x plus 7 right parenthesis space d x integral u space d v space space equals left parenthesis x squared plus 7 x minus 5 right parenthesis space left parenthesis 1 half space sin space left parenthesis 2 x right parenthesis right parenthesis minus open parentheses 1 half close parentheses open parentheses fraction numerator open parentheses negative 2 x minus 7 close parentheses over denominator 2 end fraction cos space open parentheses 2 x close parentheses plus 1 half sin open parentheses 2 x close parentheses close parentheses plus C space space space space space space space space space space space equals open square brackets left parenthesis x squared plus 7 x minus 5 right parenthesis space fraction numerator sin open parentheses 2 x close parentheses over denominator 2 end fraction plus fraction numerator open parentheses 2 x plus 7 close parentheses over denominator 4 end fraction cos open parentheses 2 x close parentheses minus 1 fourth sin open parentheses 2 x close parentheses close square brackets subscript 1 superscript 2 space space space space space space space space space space space equals open square brackets left parenthesis open parentheses 2 close parentheses squared plus 7 open parentheses 2 close parentheses minus 5 right parenthesis space fraction numerator sin open parentheses 2 open parentheses 2 close parentheses close parentheses over denominator 2 end fraction plus fraction numerator open parentheses 2 open parentheses 2 close parentheses plus 7 close parentheses over denominator 4 end fraction cos open parentheses 2 open parentheses 2 close parentheses close parentheses minus 1 fourth sin open parentheses 2 open parentheses 2 close parentheses close parentheses close square brackets space space space space space space space space space space space minus open square brackets left parenthesis open parentheses 1 close parentheses squared plus 7 open parentheses 1 close parentheses minus 5 right parenthesis space fraction numerator sin open parentheses 2 open parentheses 1 close parentheses close parentheses over denominator 2 end fraction plus fraction numerator open parentheses 2 open parentheses 1 close parentheses plus 7 close parentheses over denominator 4 end fraction cos open parentheses 2 open parentheses 1 close parentheses close parentheses minus 1 fourth sin open parentheses 2 open parentheses 1 close parentheses close parentheses close square brackets space space space space space space space space space space space equals open square brackets left parenthesis 4 plus 14 minus 5 right parenthesis space fraction numerator sin open parentheses 4 close parentheses over denominator 2 end fraction plus fraction numerator open parentheses 4 plus 7 close parentheses over denominator 4 end fraction cos open parentheses 4 close parentheses minus 1 fourth sin open parentheses 4 close parentheses close square brackets space space space space space space space space space space space minus open square brackets left parenthesis 1 plus 7 minus 5 right parenthesis space fraction numerator sin open parentheses 2 close parentheses over denominator 2 end fraction plus fraction numerator open parentheses 2 plus 7 close parentheses over denominator 4 end fraction cos open parentheses 2 close parentheses minus 1 fourth sin open parentheses 2 close parentheses close square brackets space space space space space space space space space space space equals open square brackets fraction numerator 13 space sin open parentheses 4 close parentheses over denominator 2 end fraction plus fraction numerator 11 space cos open parentheses 4 close parentheses over denominator 4 end fraction minus fraction numerator sin open parentheses 4 close parentheses over denominator 4 end fraction close square brackets minus open square brackets space fraction numerator 3 space sin open parentheses 2 close parentheses over denominator 2 end fraction plus fraction numerator 9 space cos open parentheses 2 close parentheses over denominator 4 end fraction minus fraction numerator sin space open parentheses 2 close parentheses over denominator 4 end fraction close square brackets space space space space space space space space space space space equals fraction numerator 13 space sin open parentheses 4 close parentheses over denominator 2 end fraction plus fraction numerator 11 space cos open parentheses 4 close parentheses over denominator 4 end fraction minus fraction numerator sin open parentheses 4 close parentheses over denominator 4 end fraction minus fraction numerator 3 space sin open parentheses 2 close parentheses over denominator 2 end fraction minus fraction numerator 9 space cos open parentheses 2 close parentheses over denominator 4 end fraction plus fraction numerator sin open parentheses 2 close parentheses over denominator 4 end fraction space space space space space space space space space space space equals fraction numerator 26 space sin open parentheses 4 close parentheses plus 11 space cos open parentheses 4 close parentheses minus sin open parentheses 4 close parentheses minus 6 space sin open parentheses 2 close parentheses minus 9 space cos open parentheses 2 close parentheses plus sin open parentheses 2 close parentheses over denominator 4 end fraction space space space space space space space space space space space equals fraction numerator 25 space sin open parentheses 4 close parentheses plus 11 space cos open parentheses 4 close parentheses minus 5 space sin open parentheses 2 close parentheses minus 9 space cos open parentheses 2 close parentheses over denominator 4 end fraction end style  

 

 

 

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