Question

evaluate (integral) xe^2x dx

A. 1/6x^2 e^3x+C
B. 1/2xe^2x-1/2e^2x+C
C. 1/2xe^2x-1/4e^2x+C
D. 1/2x^2-1/8e^4x+C

Evaluate (integral) e^3x cosh 2x dx

A. 1/2e^5x + 1/2e^x +C
B. 1/10e^5x + 1/2e^x + C
C. 1/4e^3x + 1/2x + C
D. 1/10e^5x + 1/5x + C

Answer 1

(1/2) (2 x e^2x dx)
let z = 2 x
then dz = 2 dx or dx = dz/2
so
(1/2) z e^z (dz/2)
(1/4) z e^z dz
but
int z e^z dz = e^z (z-1) + c
so
(1/4)e^z(z-1) + c
(1/4) e^2x (2x-1) + c
(1/2) x e^2x - (1/4) e^2x + c
so c

Answer 2

e^3x cosh 2x dx

cosh 2x = (1/2)(e^2x + e^-2x)
so
(1/2)(e^3x)(e^2x + e^-2x)dx

= (1/2)(e^5x + e^x)dx

(1/10) e^5x + (1/2) e^x + c