Integration (Cambridge (CIE) O Level Additional Maths): Exam Questions

Find the exact value of .

Given

find the exact value of .

Given that  where  , find the value of .

(i) Find , where  is a constant.

(ii) Hence find 

Find  , giving your answer in the form , where  and  are rational numbers.

A curve is such that . Given that  at the point  on the curve, find the equation of the curve.

The diagram shows part of the graphs of  and  .The graph of  meets the -axis at the point  and the two graphs intersect at the point .

Find the value of  and of .

Find the area of the shaded region.

The diagram shows the straight line  and part of the curve . The straight line intersects the -axis at the point  and intersects the curve at the point . The point  lies on the curve. The point  has coordinates (1, 0). The line  is parallel to the -axis.

Find the coordinates of each of the points  and .

Find the area of the shaded region, giving your answer in the form , where  and  are positive integers.

The diagram shows part of the curve  intersecting the straight line  at the point .
The straight line meets the -axis at the point . The point  lies on the -axis and the point  lies on the curve such that the line  has equation . Find the exact area of the shaded region, giving your answer in the form , where  and  are constants.

(i) Given that , show that .

[3]

(ii) Hence find .

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Given that , find the value of the positive constant .

Given that , where , find the exact value of , giving your answer in simplest surd form.

Find the exact value of .

Show that  can be written as 

The diagram shows part of the curve , the line  and a straight line of gradient 1. The curve intersects the -axis at the point . The line of gradient 1 passes through  and intersects the -axis at the point . Find the area of the shaded region, giving your answer in the form , where  and  are constants.

The diagram shows part of the curve  and the line , where .
The line through the maximum point of the curve, parallel to the -axis, meets the -axis at .
The curve meets the -axis at , and the line  meets the curve at the point  .

Find the area of the shaded region.

Show that  can be written as .

Hence find , giving your answer as a single logarithm and an arbitrary constant.

Given that , where , find the exact value of .

A curve is such that . This curve has a gradient of  at the point . Find the equation of this curve.

The gradient of the normal to a curve at the point  is given by 

Given that the curve passes through the point , show that its equation is .

Find, in the form , the equation of the tangent to the curve at the point where .