Homework Soursea Find a function Fx such that Fx e3xsin5x and F010 B Find
(a) Find a function F(x) such that F\'(x) = e^3x*sin(5x) and F(0)=10
(B) Find a function G(x) such that G\'(x) = 8x^3*e^2x and G(0) = 10
(B) Find a function G(x) such that G\'(x) = 8x^3*e^2x and G(0) = 10
Solution
(a) F(x) = integral of e^3xsin5x integrating wrt x F(x) = e^3x/(3^2+5^2)[3sin5x + 5cos3x] + C F(0) = 10 10 = 1/34[0 + 5] + C C = 10-34/5 C = 16/5 F(x) = e^3x/34[3sin5x + 5cos3x] + 16/5 (b) G(x) = integral of 8x^3e^2x integrating wrt x by parts G(x) = 8x^3integral of e^2x - 8integral of[d/dx(x^3) integral of e^2x] G(x) = 4x^3e^2x - 12 integral of (x^2e^2x) again apply by parts to 2nd part G(x) = 4x^3e^2x - 6x^2e^2x + 12integral of(xe^2x) again apply by parts to 3rd part G(x) = 4x^3e^2x - 6x^2e^2x + 6xe^2x - 6integral of e^2x G(x) = 4x^3e^2x - 6x^2e^2x + 6xe^2x - 3e^2x + C G(0) = 10 10 = 0-0+0-3+C C = 13 G(x) = 4x^3e^2x - 6x^2e^2x + 6xe^2x - 3e^2x + 13
