Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.

∫ 𝓍 cos²𝓍² d𝓍

검증된 단계별 안내

Step 1: Recognize that the integral involves a trigonometric function squared, specifically cos²(𝓍²). To simplify, use the trigonometric identity cos²(u) = (1 + cos(2u)) / 2, where u = 𝓍² in this case.

Step 2: Substitute the identity into the integral. The integral becomes ∫ 𝓍 * (1 + cos(2𝓍²)) / 2 d𝓍. Split the integral into two parts: ∫ 𝓍/2 d𝓍 + ∫ 𝓍 * cos(2𝓍²)/2 d𝓍.

Step 3: For the first term, ∫ 𝓍/2 d𝓍, integrate directly using the power rule for integration: ∫ 𝓍^n dx = (𝓍^(n+1)) / (n+1). This gives (𝓍² / 4).

Step 4: For the second term, ∫ 𝓍 * cos(2𝓍²)/2 d𝓍, use substitution. Let u = 2𝓍², so du = 4𝓍 d𝓍. Rewrite the integral in terms of u: (1/8) ∫ cos(u) du. The integral of cos(u) is sin(u), so this term becomes (1/8) sin(2𝓍²).

Step 5: Combine the results from both terms. The final expression for the integral is (𝓍² / 4) + (1/8) sin(2𝓍²) + C, where C is the constant of integration.

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Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals. ∫ 𝓍 cos²𝓍² d𝓍