Pure Mathematics Practice: Integration

Original practice material

These four structured questions total 42 marks. They are topic-local pure-mathematics integration practice suitable for Paper 1 or Paper 2 Section A; they are not a complete paper.

Answers: Integration Answers.

Question 1: Technique Selection

Evaluate the following integrals.

(a)

$${\int }_0^{1}x{e}^{x^2}dx.$$

[3]

(b)

$${\int }_1^{e}x\ln xdx.$$

[4]

(c)

$$\int \frac{2x}{x^2+4x+5}dx.$$

[5]

[Total: 12]

Question 2: Area Between Two Curves

The figure shows the shaded region bounded by

$$y=x(4-x)\text{and}y=x.$$

(a) Find the exact coordinates of the intersection points of the two curves. [2]

(b) Write down a definite integral for the shaded area, explaining which curve is above the other on the required interval. [3]

(c) Evaluate the shaded area. [3]

[Total: 8]

Question 3: A Linked Volume-of-Revolution Problem

The curve $C$ has equation

$$y=\sqrt{(x+1)\ln (x+1)},0\le x\le e-1.$$

(a) Using the substitution $u=x+1$, evaluate exactly

$${\int }_0^{e-1}(x+1)\ln (x+1)dx.$$

[5]

(b) The region under $C$, above the $x$-axis and between $x=0$ and $x=e-1$ is rotated through $2\pi$ radians about the $x$-axis. Hence find the exact volume generated. [3]

(c) Explain why the ordinate must be squared when forming the volume integral in part (b). [2]

[Total: 10]

Question 4: Rewriting and Signed Area

(a) Evaluate

$$\int \frac{3x+5}{x^2+2x+5}dx.$$

[5]

(b) Evaluate

$${\int }_0^{\pi /4}\frac{{\sec }^{2}x}{1+\tan x}dx.$$

[3]

(c) Let $f(x)=x^2-2x$. Find both

$${\int }_0^{3}f(x)dx$$

and the geometric area between the graph of $f$ and the $x$-axis on $[0,3]$. [4]

[Total: 12]