Solve Trigonometric Integrals with Substitution and Identities

Math Problem Statement

1. ∫ cos(2x) dx 2. ∫ sin(3x) tan(3x) dx 3. ∫ √(cos^24(4x) + 2sin^24(4x) - 1) dx

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$\int \cos(2x) \, dx$ This is a basic trigonometric integral.
$\int \sin(3x) \tan(3x) \, dx$ This integral involves a product of sine and tangent functions, which might benefit from a substitution or trigonometric identity.
$\int \sqrt{\cos^{24}(4x) + 2\sin^{24}(4x) - 1} \, dx$ This one is a bit more complex. To solve it, we'd need to simplify the expression inside the square root and possibly use trigonometric identities.

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Trigonometric Integrals
Substitution
Trigonometric Identities
Square Roots

∫ cos(ax) dx = (1/a) sin(ax)
∫ sin(ax)tan(ax) dx
Trigonometric identity: cos^2(x) + sin^2(x) = 1

Integration by Substitution
Trigonometric Identity Simplification

Grades 11-12

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Evaluate Trigonometric Integrals: Substitution and Identities