Pertanyaan

∫ e 2 x sin 3 x d x = ....

$\int e^{2 x} sin 3 x d x = ....$

Y. Endah

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

Pembahasan

Integral parsial

$table attributes columnalign right center left columnspacing 0px end attributes row u equals cell e to the power of 2 x end exponent comma space fraction numerator d u over denominator d x end fraction equals 2 e to the power of 2 x end exponent end cell row cell d v end cell equals cell sin space 3 x space d x comma space v equals integral sin space 3 x space d x comma space v equals negative 1 third cos space 3 x end cell row cell integral e to the power of 2 x end exponent space sin space 3 x space d x space end cell equals cell space u times v minus integral v space d u end cell row cell integral e to the power of 2 x end exponent space sin space 3 x space d x space end cell equals cell space minus 1 third e to the power of 2 x end exponent cos space 3 x plus 2 over 3 integral e to the power of 2 x end exponent cos space 3 x space space d x end cell row u equals cell e to the power of 2 x end exponent comma space fraction numerator d u over denominator d x end fraction equals 2 e to the power of 2 x end exponent end cell row cell d v end cell equals cell cos space 3 x space d x comma space v equals integral cos space 3 x space d x comma space v equals 1 third sin space 3 x end cell row cell integral e to the power of 2 x end exponent space sin space 3 x space d x space end cell equals cell space minus 1 third e to the power of 2 x end exponent cos space 3 x plus 2 over 3 integral e to the power of 2 x end exponent cos space 3 x space space d x end cell row cell integral e to the power of 2 x end exponent space sin space 3 x space d x end cell equals cell negative 1 third e to the power of 2 x end exponent cos space 3 x plus 2 over 3 open parentheses 1 third e to the power of 2 x end exponent sin space 3 x minus integral 2 over 3 e to the power of 2 x end exponent sin space 3 x space d x close parentheses end cell row cell integral e to the power of 2 x end exponent space sin space 3 x space d x end cell equals cell negative 1 third e to the power of 2 x end exponent cos space 3 x plus 2 over 9 e to the power of 2 x end exponent sin space 3 x minus 4 over 9 integral e to the power of 2 x end exponent sin space 3 x space d x end cell row cell 13 over 9 integral e to the power of 2 x end exponent sin space 3 x space d x end cell equals cell negative 1 third e to the power of 2 x end exponent cos space 3 x plus 2 over 9 e to the power of 2 x end exponent sin space 3 x end cell row cell integral e to the power of 2 x end exponent sin space 3 x space d x end cell equals cell 9 over 13 open parentheses negative 1 third e to the power of 2 x end exponent cos space 3 x plus 2 over 9 e to the power of 2 x end exponent sin space 3 x close parentheses end cell row blank equals cell negative 3 over 13 e to the power of 2 x end exponent cos space 3 x plus 2 over 13 e to the power of 2 x end exponent sin space 3 x plus C end cell end table$