Trigonometric Integrals

Antiderivatives that Deal with the Trigonometry

2sec tanx dx x C

2csc cotx dx x C

sec tan secx x dx x C csc cot cscx x dx x C

sin cosx dx x C cos sinx dx x C

We already know the following from Chapter 4:

But what are the integrals of tangent, secant, cosecant, and cotangent?

Example 1Find .

2csc csc cot

csc cot

x x xdx

x x

du

u

ln u C

Define u and du:

Substitute to replace EVERY x and dx:

du u 2csc cot cscx x x dx csc cotx x

csc x dx

ln csc cotx x C Substitute

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terms of x.

Integrate.

Trick:csc cot

csc csccsc cot

x xx dx x dx

x x

2csc csc cot

csc cot

x x xdx

x x

Example 2

Find

cos

sin

xdx

x

cos xdx

u1cos

cosx

xdu

u

ln u C

Define u and du:

Substitute to replace EVERY x and dx:

du u cos x dxsin xcot x dx

ln sin x C

Solve for dx

1cos x du dx

du

uSubstitute

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terms of x.

Integrate.

cos

sin

xdx

x

Integrals Involving Trigonometric Functions

tan ln cos

ln sec

x dx x C

x C

cot ln sin

ln csc

x dx x C

x C

sin cosx dx x C cos sinx dx x C

sec ln sec tanx dx x x C csc ln csc cotx dx x x C

Cotangent and Secant can be found in a similar manner as Tangent and Cosecant (respectively). The work is in the textbook.