Further Integration (Edexcel A Level Maths: Pure): Exam Questions

Express as an integral.

Hence show that

where is a constant to be found.

Given that

show that is independent of .

Given that

show that is inversely proportional to .

Given that

find the values of the constants , and

Hence, using algebraic integration, find the exact value of

giving your answer in the form where and are integers to be found.

Show that

where and are rational constants to be found.

Find

Show that

Find

Find

Use the substitution  to show that

Find

Find

Use algebraic integration to find the exact value of

Find

Show that

can be written in the form

where and are constants to be found.

Hence find

writing your answer in the form

where is a function you should find and is a constant.

The figure below shows a sketch of the curves with equations and .

The shaded region is bounded by the two curves.

Use algebraic integration to find the exact area of .

Given that

find the values of the constants and .

Hence show that if then

where is a constant.

The figure below shows a sketch of the curves with equations and .

The finite regions bounded by the two curves are shaded.

Use algebraic integration to find the exact total area of the shaded regions.

Figure 3 shows a sketch of the curve with equation .

The point lies on the curve.

The rectangle, shown shaded on Figure 3, has height and width .

Calculate

The curve with equation

where and are constants, passes through the point and has two vertical asymptotes with equations and

(i) Explain why you can deduce that

(ii) Show that

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

Show that the exact value of the area of is , where and are rational constants to be found.

Use the substitution to show that

where and are positive constants to be found.

Hence, using algebraic integration, show that

where is a rational constant to be found.

Use the substitution  to show that

where is a constant.

Use algebraic integration to show that

Find 

Use algebraic integration to show that

The figure below shows a sketch of the curve with equation  where

• the point   lies on the curve

• the shaded rectangle shown has width and height

By expressing the series as a suitable integral, show that

Use algebraic integration to show that

Use algebraic integration to show that

Use algebraic integration to show that

Use algebraic integration to show that

where , and are integers to be found and is a constant.

Show that

where is a constant.

Use algebraic integration to find

Use algebraic integration to find

Use algebraic integration to find

Use algebraic integration to find

Use algebraic integration to find

Use algebraic integration to find

Use algebraic integration to find

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

The line is the normal to at the point

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

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

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

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

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

where , and are integers to be found.

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

Solutions relying entirely on calculator technology are not acceptable.

Find the first three terms, in ascending powers of , of the binomial expansion of

writing each term in simplest form.

Using the answer to part (a) and using algebraic integration, estimate the value of

giving your answer to 4 significant figures.

Find, using algebraic integration, the exact value of

giving your answer in the form , where , and are constants to

be found.

Show that

Use the substitution to show that

where is a constant.

Find the value of

Use the substitution  to show that

Use algebraic integration to find

Prove that

where and are constants.

Use algebraic integration to show that

where is a rational number to be found.

Show that

where is a constant.

Show that

where is a constant.

The figure below shows a sketch of the curves with equations     and .

The finite regions bounded by the two curves are shaded.

Show that the -coordinates of the points of intersection are , and .

[You do not need to solve an equation in ]

Use the substitution  to show that

where is a constant.

Hence show that the exact area of the shaded regions is