22M.1.AHL.TZ2.7

pestleMathematicsAAHLPaper 122M· sl-3-5-unit-circle-definitions-of-sin-cos-tan-exact-trig-ratios-ambiguous-case-of-sine-rulesource ↗

By using the substitution $u = sec x$ or otherwise, find an expression for $\int_{0}^{\frac{π}{3}} {sec}^{n} x \tan x d x$ in terms of $n$ , where $n$ is a non-zero real number.

Markscheme / solution

METHOD 1

$u = sec x \Rightarrow d u = sec x \tan x d x$ (A1)

attempts to express the integral in terms of $u$ M1

$\int_{1}^{2} u^{n - 1} d u$ A1

$= \frac{1}{n} {[u^{n}]}_{1}^{2} (= \frac{1}{n} {[{sec}^{n} x]}_{0}^{\frac{π}{3}})$ A1

Note: Condone the absence of or incorrect limits up to this point.

$= \frac{2^{n} - 1^{n}}{n}$ M1

$= \frac{2^{n} - 1}{n}$ A1

Note: Award M1 for correct substitution of their limits for $u$ into their antiderivative for $u$ (or given limits for $x$ into their antiderivative for $x$ ).

METHOD 2

$\int {sec}^{n} x \tan x d x = \int {sec}^{n - 1} x sec x \tan x d x$ (A1)

applies integration by inspection (M1)

$= \frac{1}{n} {[{sec}^{n} x]}_{0}^{\frac{π}{3}}$ A2

Note: Award A2 if the limits are not stated.

$= \frac{1}{n} ({sec}^{n} \frac{π}{3} - {sec}^{n} 0)$ M1

Note: Award M1 for correct substitution into their antiderivative.

$= \frac{2^{n} - 1}{n}$ A1

[6 marks]

Examiners’ report

[N/A]