Pertanyaan

∫ sec n d x = n − 1 tan x sec n − 2 x + n − 1 n − 2 ∫ sec n − 2 d x ( n  = 1 )

$\int sec^{n} d x = \frac{tan x sec ^{n - 2} x}{n - 1} + \frac{n - 2}{n - 1} \int sec^{n - 2} d x (n \neq = 1)$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

terbukti bahwa

Pembahasan

Akan dibuktikan bahwa . Jika diasumsikan bahwa , maka bentuk dapat ditulis menjadi Maka dapat diaplikasikan teknik integral parsial dengan: menggunankan didapat Ingat, . Jadi: pindah ruas didapat Jadi, terbukti bahwa

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 , maka bentuk dapat ditulis menjadi

Maka dapat diaplikasikan teknik integral parsial dengan:

menggunankan didapat

Ingat, . Jadi:

pindah ruas 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