Evaluate the indefinite integral: int x * cos(x) dx.
Answer and explanation
Correct answer: A. x*sin(x) + cos(x) + C
- Step 1: Apply integration by parts: int u v' dx = uv - int u'*v dx.
- Step 2: Let u = x (so u' = 1) and v' = cos(x) (so v = sin(x)).
- Step 3: int x cos(x) dx = xsin(x) - int 1 * sin(x) dx.
- Step 4: Since int sin(x) dx = -cos(x), the expression becomes xsin(x) - (-cos(x)) + C = xsin(x) + cos(x) + C.
Answer
x*sin(x) + cos(x) + C.
Why not the others: xsin(x) - cos(x) + C has a sign error on int sin(x) dx; -xsin(x) + cos(x) + C incorrectly differentiates u; x*cos(x) - sin(x) + C fails to integrate v'.
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