Question

Find intergral (e^xcos(3x)dx

Answer 1

Answer:

Hello,

Step-by-step explanation:

Here is an other method than classical:

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