Question

Find intergral (e^xcos(3x)dx

Answer 1

Hello,

Explanation:

Here is an other method than classical:

`I=\int {e^x*cos(3x)} \, dx =e^x*(k_1/cos(3x)+k_2*sin(3x) )\\\\Let's\ derivate:\\\\e^xcos(3x)=e^x(k1*cos(3x)+k_2*sin(3x))+e^x*(k_1*(-sin(3x))*3+k_2*cos(3x)*3)\\\\By\ identification:\\\\\left\{\begin{array}{ccc}k_1+3k_2&=&1\\-3k_1+k2&=&0\\\end{array}\right.\\\\\\\left\{\begin{array}{ccc}k_1&=&(1)/(10)\\k2&=&(3)/(10)\\\end{array}\right.\\\\\\\boxed{I=e^x((1)/(10)*cos(3x)+(3)/(10)*sin(3x))}\\`