Further Integration (Edexcel International A Level (IAL) Maths: Pure 4): Exam Questions

Use a suitable substitution to find

-15 sin(5x-2) dx

Use calculus and the substitution u = x + 4 to show that

Use integration by parts to find, in terms of e, the exact value of

(5x - 4)e^3x dx

Show that

can be written in the form

Hence find

writing your answer as a single logarithm.

The diagram below shows the graph of the curve with equation y = 4 - x².

(i) Find the x-coordinates of the points where the graph of y = 4 - x² intercepts the x-axis.

(ii) The shaded region, R, is to be rotated 360° around the x-axis.

Find the volume of the shape generated.

Use a suitable substitution to find the following

Use the substitution u = 2 + In x to show that

where c is the constant of integration.

Use integration by parts to find

Show that

where c is the constant of integration.

Express

as partial fractions.

Hence, or otherwise, find

The diagram below shows a right-angled triangle with vertices at the origin, the point (h,0) and the point (h,r), where r > 0 and h > 0.

Find an equation of the line on which the hypotenuse of the right-angled triangle lies, giving your answer in the form y = f(x).

The triangle is rotated 360° about the x-axis to form a cone.

Thus use calculus to prove that the general formula for the volume, V, of a cone is

where r is the base radius of the cone and h is its perpendicular height.

The diagram below shows part of the curve C defined by the equation , where a is a positive constant. The shaded region R is bounded by the curve, the x-axis, and the lines x = 1 and x = 6.

Given that the volume of the solid formed when the region R is rotated 360° about the x-axis is cubic units, find the value of a.

Find an expression for y given that

Integrate

Use calculus and the substitution x = cos to find the exact value of

Find

Find

Find the integral

Use integration by parts to show that

where c is the constant of integration.

Sketch the region described by the following inequalities

x ≥ 2 x≤ 6 2y ≤ x + 4 y≥p, where 0<p< 2

The region described in part (a) is rotated through 360° about the x-axis.

Find the volume of the solid formed, giving your answer in terms of p.

The solid formed in part (b) will have a 'hole' in its centre.

(i) Find the volume of this 'hole', giving your answer in terms of p.

(ii) Hence show that there are no values of p in the given interval that make the volume of the solid equal to the volume of the 'hole'.

Starting with the equation of a semicircle of radius r, y = (where r > 0), use calculus to prove that the general formula for the volume, V, of a sphere of radius r is

Express in partial fractions.

Use your answer from part (a) to help find