Integration by parts

The graph of y = 4xsin 2x for 0 ≤ x ≤ π is shown below.

Show that the shaded area enclosed by the x-axis and the curve is kπ, where k is an integer to be found.

Fully justify your answer.

Show that

The region bounded by the curve

and the x-axis is shaded in the diagram below.

Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region.

Give your answer correct to three significant figures.

Show that the exact area is given by

Fully justify your answer.

Given that

find the exact value of a and the exact value of b.

Fully justify your answer.

A function f has domain and range y y e

The gradient of the curve at the point (x, y) is given by (x 1)e^x

Find an expression for f (x).

Fully justify your answer.