Consider a part of the graph of `y = xsin(x)`, as shown below.
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- Given that `int(xsin(x))\ dx = sin(x) - xcos(x) + c`,
evaluate `int_(npi)^((n + 1)pi)(xsin(x))\ dx` when `n` is a positive even integer or 0.Give your answer in simplest form. (2 marks)
- Given that `int(xsin(x))\ dx = sin(x) - xcos(x) + c`,
evaluate `int_(npi)^((n + 1)pi)(xsin(x))\ dx` when `n` is a positive odd integer.Give your answer in simplest form. (1 mark)
- Given that `int(xsin(x))\ dx = sin(x) - xcos(x) + c`,
- Find the equation of the tangent to `y = xsin(x)` at the point `(−(5pi)/2,(5pi)/2)`. (2 marks)
- The translation `T` maps the graph of `y = xsin(x)` onto the graph of `y = (3pi - x)sin(x)`, where
`qquad T: R^2 -> R^2, T([(x),(y)]) = [(x),(y)] + [(a),(0)]`
and `a` is a real constant.State the value of `a`. (1 mark)
- Let `f:[0,3pi] -> R, f(x) = (3pi - x)sin(x)` and `g:[0,3pi] -> R, g(x) = (x - 3pi)sin(x)`.
The line `l_1` is the tangent to the graph of `f` at the point `(pi/2,(5pi)/2)` and the line `l_2` is the tangent to the graph of `g` at `(pi/2,−(5pi)/2)`, as shown in the diagram below.
Find the total area of the shaded regions shown in the diagram above. (2 marks)