Evaluate the integral 4secxtanx from -pi/3 to pi/3

Question

Definite Integrals and Fundamental Theorem of Calculus

Answer 1

A function is odd if f(-x) = -f(x).

For f(x) = 4secxtanx, f(-x) = 4sec(-x)tan(-x) = 4secx (-tanx) = -4secxtanx = -f(x).

So, f(x) is an odd function.

For any odd function, ∫_{(-a to a)} f(x) dx = 0.

Therefore, ∫_{(-π/3 to π/3)} 4secxtanxdx = 0.

Answer 2

derivative of secx is secxtanx

so integral (antiderivative) of secxtanx = secx

integral of 4secxtanx = 4 times secx (ignore the constant of integration as it cancels out for a definite integral)

evaluated from -pi/3 to pi/3

= 4(sec(pi/3) - sec(-pi/3) (pi/3 =60 degrees, cos60 = 1/2, sec60 = 1/cos60 = 2)

=4( 2 - -2) = 4(2+2) = 4(4)= 16