int_{0}^{3 pi / 2} x sin x d x

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Accepted Answer

Step:1 ：Given the integral problem (int_{0}^{3 pi / 2} x sin x d x), we can solve this using the method of integration by parts.Step:2 ：The formula for integration by parts is (int udv = uv - int vdu).Step:3 ：Let&#x27;s set (u = x) and (dv = sin(x) dx).Step:4 ：We then find (du) and (v). (du) is the derivative of (x) which is (dx), and (v) is the integral of (sin(x)) which is (-cos(x)).Step:5 ：Substituting these into the formula, we get (-xcos(x) + int cos(x) dx).Step:6 ：Evaluating the integral, we get (-xcos(x) + sin(x)).Step:7 ：Evaluating this from (0) to (3 pi / 2), we get (-1).Step:8 ：So, the definite integral of (x sin x) from (0) to (3 pi / 2) is (boxed{-1}).

Answer

Step: 1 ：Given the integral problem (int_{0}^{3 pi / 2} x sin x d x), we can solve this using the method of integration by parts.Step: 2 ：The formula for integration by parts is (int udv = uv - int vdu).Step: 3 ：Let&#x27;s set (u = x) and (dv = sin(x) dx).Step: 4 ：We then find (du) and (v). (du) is the derivative of (x) which is (dx), and (v) is the integral of (sin(x)) which is (-cos(x)).Step: 5 ：Substituting these into the formula, we get (-xcos(x) + int cos(x) dx).Step: 6 ：Evaluating the integral, we get (-xcos(x) + sin(x)).Step: 7 ：Evaluating this from (0) to (3 pi / 2), we get (-1).Step: 8 ：So, the definite integral of (x sin x) from (0) to (3 pi / 2) is (boxed{-1}).

$\int_{0}^{3 π / 2} x \sin x d x$

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$\int_{0}^{3 π / 2} x \sin x d x$