Determine the area between the curve y=cos x/(4+sin x), the lines x=0 and x=pi/2 and the x axis?

We'll apply the Leibniz Newton formula to determine the
area located between the given curve and lines.

Int f(x)dx
= F(b) - F(a), where a = 0 and b = pi/2

We'll calculate the
integral of f(x) = cos x/(4+sin x):

Int cos x dx/(4+sin
x)

We'll substitute 4 + sin x =
t

We'll differentiate both
sides:

cos xdx = dt

We'll
re-write the integral:

Int cos x dx/(4+sin x) = Int dt/t =
ln |t| + C

We'll determine F(b) -
F(a):

F(pi/2) = ln (4 + sin pi/2) = ln (4 + 1) = ln
5

F(0) = ln (4 + sin 0) = ln (4 + 0) = ln
4

Int f(x)dx = F(pi/2) -
F(0)

Int f(x)dx = ln 5 - ln
4

Int f(x)dx = ln
(5/4)

The area under the curve is ln (5/4)
square units.