Find \(\int {x{{\sec }^2}x{\text{d}}x} \).

Determine the value of m if \(\int_0^m {x{{\sec }^2}x{\text{d}}x = 0.5} \), where m > 0.

\(\int {x{{\sec }^2}x{\text{d}}x}  = x\tan x - \int {1 \times \tan x{\text{d}}x} \)     M1A1

\( = x\tan x + \ln \left| {\cos x} \right|( + c){\text{ }}\left( { = x\tan x - \ln \left| {\sec x} \right|( + c)} \right)\)     M1A1

[4 marks]

attempting to solve an appropriate equation eg \(m\tan m + \ln (\cos m) = 0.5\)     (M1)

m = 0.822     A1

Note: Award A1 if m = 0.822 is specified with other positive solutions.

[2 marks]

In part (a), a large number of candidates were able to use integration by parts correctly but were unable to use integration by substitution to then find the indefinite integral of tan x. In part (b), a large number of candidates attempted to solve the equation without direct use of a GDC’s numerical solve command. Some candidates stated more than one solution for m and some specified m correct to two significant figures only.

In part (a), a large number of candidates were able to use integration by parts correctly but were unable to use integration by substitution to then find the indefinite integral of tan x. In part (b), a large number of candidates attempted to solve the equation without direct use of a GDC’s numerical solve command. Some candidates stated more than one solution for m and some specified m correct to two significant figures only.