Unit 8 - Objective 4 - Integrals of Trig Functions



From the six derivative formulas for trig functions we get the following integral formulas.




We also have the following formulas for integrals of the remaining trig functions which are not so obvious.





Use these integral formulas and any formulas from previous worksheets to do the following examples. For these integrals above, u is the angle.

Problem 1:

sin 3x dx           angle = u = 3x, du = 3 dx, "fix" the 3
sin 3x dx = (1/3) sin(3x) (3)dx = (1/3) sin (u) du

= (1/3) (-cos (u)) + C

= -(1/3) cos 3x + C


Problem 2:

cos 8x dx  =  (1/8) cos 8x (8) dx = (1/8)sin 8x + C


Problem 3:

This is a fraction with the denominator to the first power. Try the log integral.

u = x + sin x, du = (1 + cos x) dx


Problem 4:

This does not match any of the formulas above. It is a fraction with the denominator to the first power, but the numerator is not the derivative of the denominator. Use trig identities to rewrite the integrand.

= 5 csc 3x dx       constants factor out

= 5 csc 3x dx        "fix" du = 3 dx

= 5 (1/3) csc 3x (3) dx

= 5 (1/3) ln |csc 3x - cot 3x| + C

= (5/3) ln |csc 3x - cot 3x| + C



Problem 5:

x sec x² tan x² dx
u = x²,  du = 2x dx,   "fix" the 2
x sec x² tan x² dx = (1/2) (2) x sec x² tan x² dx

= (1/2) sec x² + C


Problem 6:





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