integral cos^2 3x . sin^4 3x dx
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cos^2(3x)sin^4(3x)
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Step-by-step solution
Cos^2 (3x)sin^4(3x)dx
To evaluate the integral ∫cos^2(3x) * sin^4(3x) dx, we can use a combination of trigonometric identities and the power-reducing formula.
Step 1: Rewrite the integral using the power-reducing formula for cosine: cos^2(θ) = (1 + cos(2θ)) / 2.
∫(1 + cos(2*3x))/2 * sin^4(3x) dx
Step 2: Expand sin^4(3x) using an identity. Since sin^2(θ) = (1 - cos(2θ))/2, we can rewrite sin^4(θ) as (sin^2(θ))^2.
∫(1 + cos(2*3x))/2 * (sin^2(3x))^2 dx
Step 3: Apply the power-reducing formula for sine: sin^2(θ) = (1 - cos(2θ))/2.
∫(1 + cos(2*3x))/2 * ((1 - cos(2*3x))/2)^2 dx
Step 4: Simplify the expression.
∫(1 + cos(6x))/2 * (1 - cos(6x))^2/4 dx
Step 5: Expand the squared expression.
∫(1 + cos(6x))/2 * (1 - 2cos(6x) + cos^2(6x))/4 dx
Step 6: Distribute and simplify.
∫(1 - 2cos(6x) + cos^2(6x) + cos(6x) - 2cos^2(6x) + cos^3(6x))/8 dx
Step 7: Combine like terms.
∫(1 + cos(6x) - cos^2(6x) + cos^3(6x))/8 dx
Step 8: Evaluate each term separately.
∫(1/8 + cos(6x)/8 - cos^2(6x)/8 + cos^3(6x)/8) dx
Step 9: Integrate each term.
(1/8)x + (1/48)sin(6x) - (1/8)(1/2)x - (1/24)sin(2*6x)/2 - (1/8)(1/3)sin^3(6x)/3 + C
Simplifying further:
(1/8)x + (1/48)sin(6x) - (1/16)x - (1/48)sin(12x) - (1/24)(1/3)sin^3(6x)/3 + C
Finally:
(1/16)x + (1/48)sin(6x) - (1/48)sin(12x) - (1/72)sin^3(6x) + C
So, the answer to the integral is (1/16)x + (1/48)sin(6x) - (1/48)sin(12x) - (1/72)sin^3(6x) + C.