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Integration Trigonometric Functions

(1) S cos2xsin2xdx

(2) S tan2xsecxdx

(3) S sinxcosxdx using four different methods

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(1)

∫cos2xsin2xdx

Use the fact that sin2x = 2sinxcosx:

∫cos2xsin2xdx = ∫cos2x(2sinxcosx)dx

Move the 2 outside the integral and combine the cosx terms:

2∫cos3xsinxdx

The derivative of cosx is -sinx, so use u substitution (u = cosx, du = -sinxdx):

-2∫u3du

Integrate:

-2(u4/4) + c

Simplify and plug cosx back in for u:

-cos4x + c
2

(2)

∫tan2xsecxdx

We know that tan2x = sec2 - 1:

∫tan2xsecxdx = ...

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