Integration by parts

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 5 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 5, is bounded by

• the curve C

• the line with equation x = 1

• the x-axis

Find the exact area of R, giving your answer in the form

where A and B are rational numbers to be found.

Figure 2 shows a sketch of part of the curve with equation

The finite region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 4

The table below shows corresponding values of x and y, with the values of y given to 4 decimal places.

Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of R, giving your answer to 3 significant figures.

Use algebraic integration to find the exact area of R, giving your answer in the form

where a, b and c are integers to be found.

Figure 2 shows a sketch of part of the curve with equation

The finite region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 4 The table below shows corresponding values of x and y, with the values of y given to 4 decimal places.

Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of R, giving your answer to 3 significant figures.

Use algebraic integration to find the exact area of R, giving your answer in the form

where a, b and c are integers to be found.

Show that

Figure 2 shows a sketch of part of the curve C with equation y xlnx, x 0

The line l is the normal to C at the point P(e, e)

The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the x-axis.

Show that the exact area of R is Ae² B where A and B are rational numbers to be found.

Find, in its simplest form,

Hence find

Use integration by parts to find

giving your answer in the form a + b ln 2, where a and b are constants to be found.

Figure 1 shows a sketch of part of the curve with equation y = x²ln x, x 1 The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 2 The table below shows corresponding values of x and y for y = x²ln x

Complete the table above, giving the missing value of y to 4 decimal places.

Use the trapezium rule with all the values of y in the completed table to obtain an estimate for the area of R, giving your answer to 3 decimal places.

Use integration to find the exact value for the area of R.

The curve meets the x-axis at the origin O and cuts the x-axis at the point A.

Find, in terms of ln2, the x coordinate of the point A.

Find

The finite region R, shown shaded in Figure 1, is bounded by the x-axis and the curve with equation

Find, by integration, the exact value for the area of R. Give your answer in terms of ln2

Figure 1 shows a sketch of part of the curve with equation

The finite region R, shown shaded in Figure 1, is bounded by the curve, the x- axis and the y- axis. The table below shows corresponding values of x and y for y = (2 – x)e^2x

Use the trapezium rule with all the values of y in the table, to obtain an approximation for the area of R, giving your answer to 2 decimal places.

Explain how the trapezium rule can be used to give a more accurate approximation for the area of R.

Use calculus, showing each step in your working, to obtain an exact value for the area of R. Give your answer in its simplest form.

3 decimal places.

Use the trapezium rule with all the values of x in the completed table to obtain an estimate for the area of the region R, giving your to 2 decimal places. (3)

Use calculus to find the exact value for the area of R. (6)

Find the difference between the values obtained in part (b) and part (c), giving your to 2 decimal places. (1)

Use integration to find

Hence calculate

Figure 3 shows a sketch of part of the curve with equation y = ln 2x. The finite region R, shown shaded in Figure 3, is bounded by the curve, the x-axis and the lines x = 1 and x 4

Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of R, giving your answer to 2 decimal places.

Hence find the exact area of R, giving your answer in the form a ln 2 b, where a and b are exact constants.

Figure 2 shows a sketch of the curve with equation y = x³ ln (x² + 2), x0. The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x = ˲ 2. The table below shows corresponding values of x and y for y = x³ ln (x² + 2).

Complete the table above giving the missing values of y to 4 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places.

Use the substitution u = x²+ 2 to show that the area of R is

Hence, or otherwise, find the exact area of R.

₂₂ f(θ) = 4 cos θ – 3 sin θ

Figure 1 shows a sketch of the curve with equation y = x 1n x, x 1. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 4. The table shows corresponding values of x and y for y = x 1n x.

Complete the table with the values of y corresponding to x = 2 and x = 2.5, giving your answers to 3 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places.

Hence find the exact area of R, giving your answer in the form a ln 2 b), where a and b are integers.

Use the substitution u^x to show that = 1 + e

where k is a constant.

2._x (a) Use integration by parts to find. x e dx

Find the exact value of sin²x dx.