Integration by substitution

Use the substitution u = e + 1 to show that

Solve the differential equation

given that y = π when x = 0

Write your answer in the form

where A and B are constants to be found.

Use a suitable substitution to show that

can be written as

where a is a constant to be found.

Hence, or otherwise, show that

A graph has the equation

A student uses four rectangles to approximate the area under the graph between the lines x = 0 and x = 4

The rectangles are all the same width.

All the rectangles are drawn under the curve as shown in the diagram below.

The total area of the four rectangles is A

The student decides to improve their approximation by increasing the number of rectangles used.

Explain why the value of the student’s improved approximation will be greater than A, but less than

Use the substitution u x⁵ 2 to show that

Given that

Express y in terms of sin .

Hence, prove that

Show that

Use the substitution

to show that

can be written as

where k is a constant to be found.

Hence, show

where c is a constant.

The curve C is defined for t 0 by the parametric equations

2 2 3

C is shown in the diagram below.

Find the gradient of C at the point where it intersects the positive x-axis.

The area A enclosed between C and the x-axis is given by

Find the value of b.

Use the substitution y 4t² t³ to show that

2 3 4

Find the value of A.

Use integration by substitution to show that

Fully justify your answer.

Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into n trapezia.

When n 4

State the number of ordinates that Caroline uses.

Calculate the area that Caroline should obtain using this method.

Give your answer correct to two decimal places.

Show that the exact area of A is

Fully justify your answer.

Explain what would happen to Caroline’s answer to part (a)(ii) as n