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∫sin⁡(8x) dx9+sin⁡4(4x)\int \frac{\sin(8x)\,dx}{9+\sin^{4}(4x)}

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Full solution

Q. ∫sin⁡(8x) dx9+sin⁡4(4x)\int \frac{\sin(8x)\,dx}{9+\sin^{4}(4x)}
  1. Simplify Integral: Simplify the integral if possible.\newlineThe integral given is ∫sin⁡(8x) dx9+sin⁡4(4x)\int \frac{\sin(8x)\,dx}{9 + \sin^4(4x)}. There is no straightforward simplification or standard integral formula that applies directly to this integral. We will need to consider a substitution or a trigonometric identity to simplify the integrand.
  2. Trig Identity/Substitution: Look for a trigonometric identity or substitution that can simplify the integrand.\newlineNotice that sin⁡2(4x)\sin^2(4x) can be written as (1−cos⁡(8x))/2(1 - \cos(8x))/2 using the double-angle formula for cosine: cos⁡(2θ)=1−2sin⁡2(θ)\cos(2\theta) = 1 - 2\sin^2(\theta). However, we have sin⁡4(4x)\sin^4(4x) in the denominator, which is (sin⁡2(4x))2(\sin^2(4x))^2. This suggests that we might need to use a different approach since the numerator does not simplify easily with this identity.
  3. Consider Substitution: Consider a substitution that might simplify the integral.\newlineLet's try a substitution involving the argument of the sine function in the numerator. Let u=4xu = 4x, which implies that du=4dxdu = 4dx or dx=du4dx = \frac{du}{4}. We will substitute this into the integral and adjust the limits accordingly.
  4. Perform Substitution: Perform the substitution.\newlineSubstituting u=4xu = 4x and dx=du4dx = \frac{du}{4} into the integral, we get:\newline∫sin⁡(8x)dx9+sin⁡4(4x)=∫sin⁡(2u)⋅(du4)9+sin⁡4(u)\int \frac{\sin(8x)dx}{9 + \sin^4(4x)} = \int \frac{\sin(2u) \cdot \left(\frac{du}{4}\right)}{9 + \sin^4(u)}\newlineThis simplifies to 14∫sin⁡(2u)du9+sin⁡4(u)\frac{1}{4}\int \frac{\sin(2u)du}{9 + \sin^4(u)}
  5. Further Simplifications: Look for further simplifications or applicable integration techniques.\newlineThe integral (1/4)∫(sin⁡(2u) du)/(9+sin⁡4(u))(1/4)\int(\sin(2u)\,du) / (9 + \sin^4(u)) still does not have an obvious solution. We might need to use a more advanced integration technique such as partial fractions, trigonometric identities, or a numerical method. However, none of these methods seem to apply directly, and the integral may not have a closed-form solution expressible in terms of elementary functions.
  6. Determine Solution: Determine if the integral can be solved with elementary functions.\newlineAfter considering various methods, it appears that the integral does not have a solution in terms of elementary functions. Therefore, we cannot provide an exact, simplified answer to this integral using standard calculus techniques.

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