Roboguru

Pertanyaan

begin mathsize 14px style integral sec to the power of n space d x equals fraction numerator tan space x space sec to the power of n minus 2 end exponent space x over denominator n minus 1 end fraction plus fraction numerator n minus 2 over denominator n minus 1 end fraction integral sec to the power of n minus 2 end exponent space d x space left parenthesis n not equal to 1 right parenthesis end style

Pembahasan Soal:

Akan dibuktikan bahwa begin mathsize 14px style integral s e c to the power of n space x space d x equals fraction numerator tan space x space s e c to the power of n minus 2 end exponent space x over denominator n minus 1 end fraction plus fraction numerator n minus 2 over denominator n minus 1 end fraction integral s e c to the power of n minus 2 end exponent space x space d x space left parenthesis n not equal to 1 right parenthesis end style.
 

Jika diasumsikan bahwa begin mathsize 14px style n greater than 2 end style, maka bentuk begin mathsize 14px style integral sec to the power of n space x space d x end style dapat ditulis menjadi
 

begin mathsize 14px style integral sec to the power of n space x space d x equals integral sec to the power of n minus 2 end exponent space x space sec space x space d x end style
 

Maka dapat diaplikasikan teknik integral parsial dengan:
 

begin mathsize 14px style open curly brackets table attributes columnalign left end attributes row cell u equals s e c to the power of n minus 2 end exponent space x rightwards double arrow d u equals left parenthesis n minus 2 right parenthesis space s e c to the power of n minus 3 end exponent x space s e c space x space tan space x space d x end cell row cell d v equals s e c squared x space d x rightwards double arrow v equals tan space x end cell end table close end style
 

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

begin mathsize 14px style integral s e c to the power of n minus 2 end exponent space x space s e c space x space d x equals left parenthesis s e c to the power of n minus 2 end exponent space x right parenthesis left parenthesis space tan space x right parenthesis minus integral tan x space left parenthesis left parenthesis n minus 2 right parenthesis space s e c to the power of n minus 3 end exponent space x space s e c space x space tan space x right parenthesis space d x end style
 

menggunankan begin mathsize 14px style 1 plus tan squared space x equals space s e c squared space x end style didapat
 

begin mathsize 14px style integral s e c to the power of n minus 2 end exponent space x space s e c space d x equals tan space x space s e c to the power of n minus 2 end exponent space x space minus left parenthesis n minus 2 right parenthesis integral left parenthesis s e c squared space x minus 1 right parenthesis space s e c to the power of n minus 2 end exponent space x space d x equals tan space x space s e c to the power of n minus 2 end exponent space x space minus left parenthesis n minus 2 right parenthesis integral s e c to the power of n space x minus s e c to the power of n minus 2 end exponent space x space d x equals tan space x space s e c to the power of n minus 2 end exponent space x space minus left parenthesis n minus 2 right parenthesis integral s e c to the power of n space x space d x plus left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x end style
 

Ingat, begin mathsize 14px style integral s e c to the power of n space x space d x equals integral s e c to the power of n minus 2 end exponent space x space s e c space x space d x end style. Jadi:
 

begin mathsize 14px style integral s e c to the power of n space d x equals tan space x space s e c to the power of n minus 2 end exponent space x minus left parenthesis n minus 2 right parenthesis integral s e c to the power of n space x space d x plus left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x end style
 

pindah ruas begin mathsize 14px style left parenthesis n minus 2 right parenthesis integral s e c to the power of n space x space d x end style didapat
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral s e c to the power of n space d x plus open parentheses n minus 2 close parentheses integral s e c to the power of n space x space d x end cell equals cell tan space x space s e c to the power of n minus 2 end exponent space x plus left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x end cell row cell integral s e c to the power of n space x space d x times left parenthesis 1 plus left parenthesis n minus 2 right parenthesis right parenthesis end cell equals cell tan space x space s e c to the power of n minus 2 end exponent space x plus left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x end cell row cell integral s e c to the power of n space x space d x times left parenthesis n minus 1 right parenthesis end cell equals cell tan space x space s e c to the power of n minus 2 end exponent space x plus left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x end cell row cell integral s e c to the power of n space x space d x end cell equals cell fraction numerator tan space x space s e c to the power of n minus 2 end exponent space x plus left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x over denominator left parenthesis n minus 1 right parenthesis end fraction end cell row cell integral s e c to the power of n space x space d x end cell equals cell fraction numerator tan space x space s e c to the power of n minus 2 end exponent space x over denominator n minus 1 end fraction plus fraction numerator left parenthesis n minus 2 right parenthesis integral s e c to the power of n minus 2 end exponent space x space d x over denominator n minus 1 end fraction end cell row cell integral s e c to the power of n space x space d x end cell equals cell fraction numerator tan space x space s e c to the power of n minus 2 end exponent space x over denominator n minus 1 end fraction plus fraction numerator left parenthesis n minus 2 right parenthesis over denominator n minus 1 end fraction integral s e c to the power of n minus 2 end exponent space x space d x space left parenthesis n not equal to 1 right parenthesis end cell end table end style
 

Jadi, terbukti bahwa undefined

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 12 Maret 2021

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