Again More Trig Integrals

In a Nut Shell: Sometimes you just need to express trig functions in terms of their

basic definition. i.e. Tangent is simply sine divided by cosine.

Example 5: ∫ tan x dx = ∫ [sin x / cos x] dx

u = cos x du = - sin x dx

∫ tan x dx = - ∫ du /u which is a standard integral, (ln u)

In similar manner ∫ cot x dx = ∫ [cos x / sin x] dx

u = sin x du = cos x dx

∫ cot x dx = ∫ du /u which is a standard integral, (ln u)

In a Nut Shell: Sometimes you may need special tricks such as multiplying and

dividing by the same function followed by a substitution.

Example 6: ∫ sec x dx

Multiply and divide sec x by (sec x + tanx)

and let u = sec x + tanx, du = (sec x tan x + sec²x ) dx

So integral becomes ∫ du / u = ln |u| + C

∫ sec x dx = ln | sec x + tan x | + C

Use similar strategy for ∫ csc x dx (don’t forget – sign)

∫ csc x dx = - ln | csc x + cot x | + C

Return to Notes for Calculus 2

Copyright © 2017 Richard C. Coddington

All rights reserved.