Evaluate the indefinite integral

Question

∫e^(4x)*sin(7x)dx?

Answer 1

Let y = ʃe^4x  sin(7x)dx = ʃsin(7x) e^4x dx

The integration product formula

ʃuv = uʃv - ʃu'ʃv

                          = sin7x ʃe^4x - ʃ7cos7xʃe^4xxdx

                          = sin7x×e^4x / 4 - ʃ7cos7xe^4x / 4dx

                         = sin7x×e^4x / 4 - 7 / 4[cos7x × e^4x  / 4 - ʃ-7sin7x×e^4x / 4 dx]

                    y = sin7x ×e^4x / 4 - 7 / 16cos7x×e^4x -49 / 16[ʃsin7x×e^4x dx]

                   y = sin7x ×e^4x / 4 - 7 / 16cos7x×e^4x - 49 / 16[y]

                  y = sin7x ×e^4x / 4 - 7 / 16cos7x×e^4x - 49y / 16

                 y + 49y / 16 = sin7x ×e^4x / 4 - 7 / 16cos7x×e^4x

                      65y / 16 = sin7x ×e^4x / 4 - 7 / 16cos7x×e^4x

                          y = 16 / 65[sin7x ×e^4x / 4 - 7 / 16cos7x×e^4x]

                          y = 1 / 65[4sin7x ×e^4x  - 7 cos7x×e^4x]

                          y = e^4x / 65[4sin7x - 7cos7x].