The integral to be evaluated is ∫ sin²(x) cos³(x) dx. The correct answer is 1/5 - sin³(x) + C, where C is the constant of integration.
To evaluate the integral ∫ sin²(x) cos³(x) dx, we can use trigonometric identities to simplify the integrand. Using the identity sin²(x) = 1/2 - (1/2) cos(2x) and cos³(x) = cos(x) cos²(x), we can rewrite the integral as:
∫ (1/2 - (1/2) cos(2x)) (cos(x) cos²(x)) dx
Expanding the integrand and distributing, we have:
(1/2) ∫ cos(x) cos²(x) dx - (1/2) ∫ cos(2x) cos²(x) dx
Now, we can use integration formulas to evaluate each integral separately. For the first integral, we can use the substitution u = sin(x) to simplify it:
(1/2) ∫ cos(x) (1 - sin²(x)) dx
Letting du = cos(x) dx, the integral becomes:
(1/2) ∫ (1 - u²) du = (1/2) (u - (1/3)u³) + C = (1/2) (sin(x) - (1/3)sin³(x)) + C
For the second integral, we can use the double-angle identity cos(2x) = 2cos²(x) - 1 to simplify it:
-(1/2) ∫ (2cos²(x) - 1) cos²(x) dx
Expanding and integrating, we have:
-(1/2) [2 ∫ cos⁴(x) dx - ∫ cos²(x) dx] = -(1/2) [(2/5) cos⁵(x) - (1/3)cos³(x)] + C
Combining the results from both integrals, we get:
∫ sin²(x) cos³(x) dx = (1/2) (sin(x) - (1/3)sin³(x)) - (1/2) [(2/5) cos⁵(x) - (1/3)cos³(x)] + C
Simplifying further, we obtain:
(1/2) (sin(x) - (1/3)sin³(x) - (4/5) cos⁵(x) + (1/3)cos³(x)) + C
This can be further simplified to:
1/5 - sin³(x) + C
Therefore, the correct answer to the integral ∫ sin²(x) cos³(x) dx is 1/5 - sin³(x) + C, where C is the constant of integration.
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