Evaluate the definite integral: integral from 0 to pi/2 of [cos(x) * sin^3(x)] dx.
Answer and explanation
Correct answer: B. 1/4
- Step 1: Use substitution: let u = sin(x), then du = cos(x) dx.
- Step 2: Determine transformed limits: when x = 0, u = 0; when x = pi/2, u = 1.
- Step 3: Integrate u^3 du: integral from 0 to 1 of u^3 du = [u^4 / 4] from 0 to 1 = 1/4 - 0 = 1/4.
Answer
1/4.
Why not the others: '1/3' uses the power 3 in the denominator instead of 4; '1/2' integrates sin(x)cos(x) instead of sin^3(x); '1' evaluates the boundary value without integrating.
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