Integration by substitution

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

Figure 5 shows a sketch of the curve C with equation

The region R, shown shaded in Figure 5, ounded by C, the line with equation x = 1,

Use the substitution x = 2 sinu to show that the area of R is given by

where a, b and k are constants to be found.

Hence, using algebraic integration, find the exact area of R. Give your answer in simplest form.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve with equation y = f (x), where

The curve

• intersects the x ‑axis at –1 and 1

• has minimum turning points at P and Q

as shown in Figure 5.

Use calculus to find the exact coordinates of P.

Using the substitution x = tanθ show that

where α and β are constants to be found. The finite region R, shown shaded in Figure 5, is bounded by the x‑axis and the curve.

Use algebraic integration to find the area of R.

Given that a is a positive constant, use the substitutionx = asin² θto show that

Hence use algebraic integration to show that

where k is a constant to be found.

where p and q are constants to be found.

Hence show that

where A and B are constants to be found.

where p and q are constants to be found.

Hence show that

where A and B are constants to be found.

Use the substitution x = u² + 1 to show that

where p and q are positive constants to be found.

Hence, using algebraic integration, show that

where a is a rational constant to be found.

where k is a constant

A team of scientists is studying a species of slow growing tree. The rate of change in height of a tree in this species is modelled by the differential equation

where h is the height in metres and t is the time, measured in years, after the tree is planted.

Find, according to the model, the range in heights of trees in this species.

One of these trees is one metre high when it is first planted. According to the model,

calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.

Show that

Figure 1 shows a sketch of the curve with equation

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

The table below shows corresponding values of x and y for y = sin 2x sin x௘ ZLWK WKH YDOXHV of y given to 5 significant figures where appropriate.

Use the trapezium rule, with all the values of y in the table, to find an estimate for the area of R, giving your answer to 4 significant figures.

where a and b are constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Given that y 0, find

Use the substitution x = 4sin² to show that

Hence use integration to find

giving your answer in the form a + b, where a and b are exact constants.

The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis, and the y-axis.

Use the substitution x = 1 + 2sinθ to show that

where k is a constant to be determined.

Hence find, by integration, the exact area of R.

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The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, and the lines with equations x = 1 and x = 4

Complete the table above by giving the missing value of y to 5 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to find an estimate for the area of R, giving your answer to 4 decimal places.

By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of R.

Use the substitution u = ¥ x, or otherwise, to find the exact value of

value of

Use the substitution x = u², u > 0, to show that

Hence show that

where a and b are integers to be determined.

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

Complete the table with the value of y corresponding to x = 3, giving your answer to 4 decimal places.

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

Use the substitution u = 1 + x, to find, by integrating, the exact area of R.

Figure 2 shows a sketch of part of the curve C with parametric equations

The curve crosses the y-axis at the point A and crosses the x-axis at the point B.

Show that A has coordinates (0, 3).

Find the x coordinate of the point B.

Find an equation of the normal to C at the point A.

The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and the x-axis.

Use integration to find the exact area of R.

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.

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

Using the substitution x = u 4² +1, or otherwise, and integrating, find the exact value of I.

Using the substitution u = cos x +1, or otherwise, show that

Using the substitution x = 2 cos u, or otherwise, find the exact value of

The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution.

Using your answer to part (a), find the exact volume of the solid of revolution formed.

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

where k is a constant.

Figure 3 shows the curve C with parametric equations

The point P lies on C and has coordinates (4, 2√3).

Find the value of t at the point P.

The line l is a normal to C at P.

Show that an equation for l is y = –x√3 + 6√3.

The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3.

Show that the area of R is given by the integral.²64 sin² t cost dt

Use this integral to find the area of R, giving your answer in the form a + b√3, where a and b are constants to be determined.