Evaluate integraldisplay π 2 x sin2 x dx a π 4 b π

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5. Evaluate integraldisplay π/ 2 0 x sin(2 x ) dx . (a) π/ 4 (b) π/ 3 (c) π/ 2 (d) π (e) 2 π (f) 3 π 6. Evaluate integraldisplay 1 0 sin 3 ( π 2 x ) dx . 7. Suppose that f ( x ) is a function such that f ′′ ( x ) = cos( x 3 ). The trapezoidal rule is then used to approximate the integral integraldisplay 1 0 f ( x ) dx , using ten subintervals of equal length. What is the strongest statement that can be made about the size of the error E , based just on the general error bound for approximations via the trapezoidal rule? 8. The improper integral integraldisplay 1 0 1 (2 x - 1) 1 3 dx 1
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9. Find the arc length of the graph of y = x 3 3 + 1 4 x between x = 1 and x = 2. [Note: It may be helpful to use identities like ( x 2 + 1 4 x 2 ) 2 = x 4 + 1 2 + 1 16 x 4 .] (a) 0 (b) 59 / 24 (c) 8 27 (10 10 - 1) (d) π ln(2) (e) 3 8 + ln(2) (f) It is divergent.
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