Integrals of Trigonometric Functions

Key Questions

  • What is the antiderivative of tan(x)?

    Recall:

    int{g'(x)}/{g(x)}dx=ln|g(x)|+C

    (You can verify this by substitution u=g(x).)

    Now, let us look at the posted antiderivative.

    By the trig identity tan x={sin x}/{cos x},

    int tan x dx=int{sin x}/{cos x}dx

    by rewriting it a bit further to fit the form above,

    =-int{-sin x}/{cos x}dx

    by the formula above,

    =-ln|cos x|+C

    or by rln x=lnx^r,

    =ln|cos x|^{-1}+C=ln|sec x|+C

    I hope that this was helpful.

    Wataru · 1 · Oct 16 2014
  • What is the antiderivative of sec^2(x)?

    Since

    (tanx)'=sec^2x,

    we have

    int sec^2x dx=tan x +C.

    I hope that this was helpful.

    Wataru · 3 · Oct 16 2014
  • What are the antiderivatives of sec(x), csc(x) and cot(x)?

    Since

    (ln|secx+tanx|)'={secxtanx+sec^2x}/{sec x+tanx}=secx,

    we have

    int secx dx=ln|secx+tanx|+C

    Since

    (-ln|cscx+cotx|)'=-{-cscxcotx-csc^2x}/{cscx+cotx}=cscx,

    we have

    int cscx dx=-ln|cscx+cotx|+C

    int cotx dx=int{cosx}/{sinx}dx=ln|sinx|+C

    I hope that this was helpful.

    Wataru · · Oct 16 2014
  • What are the antiderivatives of sin(x) and cos(x)?

    Since

    (sinx)'=cosx,

    we have

    int cosx dx=sinx +C.

    Since

    (-cosx)'=sinx,

    we have

    int sinx dx=-cosx+C.

    I hope that this was helpful.

    Wataru · · Oct 16 2014

Questions

Introduction to Integration