Math Problem Statement

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

Solution

It looks like you've uploaded an image containing three integrals. Let's go through each of them one by one:

  1. cos(2x)dx\int \cos(2x) \, dx This is a basic trigonometric integral.

  2. sin(3x)tan(3x)dx\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.

  3. cos24(4x)+2sin24(4x)1dx\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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Math Problem Analysis

Mathematical Concepts

Trigonometric Integrals
Substitution
Trigonometric Identities
Square Roots

Formulas

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

Theorems

Integration by Substitution
Trigonometric Identity Simplification

Suitable Grade Level

Grades 11-12