Evaluate the following integrals using properties of integra

Evaluate the following integrals using properties of integrals or the substitution method:

(a) /2 integral0 (sin3 x cos x dx)

b) 2 integral1 ((sqrt 2x + 5) dx )

(c) integral (sin x cos x + (5^x  ) + 9 sec^2 x) dx

(d) integral ((5)/( x ^2 + 2x + 1) + 3 sec x tan x) dx

Solution

a. pi/2 integral 0(sin3xcosx)dx

sin 3x=3sinx -4sin³x

     pi/2 integral 0(3sinx - 4sin³x)cosx dx

let sinx =t

cos x dx=dt

whe x=pi/2 ,t=1 and when x=0,t=0

1 integral0 (3t-t³)dt = (3t²/2    - 4t⁴/4)₀¹= (3(1)²/2 - 1⁴)-(3(0)²/2 - (0)⁴ )   

                                                        = 3/2 -1=1/2