IB Questionbank

Loading [MathJax]/jax/element/mml/optable/BasicLatin.js

Updates to Questionbank

User interface language: English | Español

Date	May 2022	Marks available	6	Reference code	22M.1.AHL.TZ2.7
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Find	Question number	7	Adapted from	N/A

Question

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

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]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule

Show 28 related questions

16N.2.AHL.TZ0.H_7b:
Find the difference between the areas of the two possible triangles ABC.
19N.1.AHL.TZ0.H_9b:
Show that ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 75^\circ$ .
19M.2.SL.TZ2.S_3a:
Find BD.
19M.2.SL.TZ2.S_3b:
The area of triangle ABD is 18.5 cm². Find the possible values of θ.
19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
22M.1.SL.TZ1.4b:
Solve the equation $(f \circ g) (x) = 2 \cos 2 x$ where $0 \leq x \leq π$ .
22M.1.SL.TZ1.9b.i:
Show that $\int_{0}^{m} f (x) d x = \frac{32}{7} \sin^{7} m$ , where $m$ is a positive real constant.
22M.1.SL.TZ1.9b.ii:
It is given that $\int_{m}^{\frac{π}{2}} f (x) d x = \frac{127}{28}$ , where $0 \leq m \leq \frac{π}{2}$ . Find the value of $m$ .
22M.1.AHL.TZ1.9b:
Hence, given that $arg (z_{1} z_{2}) = \frac{π}{4}$ , find the value of $b$ .
22M.2.AHL.TZ2.7b:
Using l’Hôpital’s rule, show algebraically that the value of the limit is $- \frac{1}{4}$ .
18M.2.SL.TZ1.S_6:
Triangle ABC has a = 8.1 cm, b = 12.3 cm and area 15 cm². Find the largest possible perimeter of triangle ABC.
18N.2.SL.TZ0.S_7a:
Let SR = $x$ . Use the cosine rule to show that ${x^2} - \left( {76\,{\text{cos}}\,43^\circ } \right)x + 420 = 0$ .
16N.1.AHL.TZ0.H_13d:
Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval $0 < x < \pi$ .
19M.1.AHL.TZ1.H_4a:
Show that ${\text{sin}}\,\theta = \frac{{\sqrt {15} }}{4}$ .
19N.1.AHL.TZ0.H_9c.i:
By considering triangle ${\text{ABD}}$ , show that ${\text{B}}{{\text{D}}^2} = 5{r^2} - 2{r^2}q\sqrt 6$ .
19N.1.AHL.TZ0.H_9d:
Use your answers to part (c) to show that ${\text{cos}}\,75^\circ = \frac{1}{{\sqrt 6 + \sqrt 2 }}$ .
22M.1.SL.TZ2.5:
Find the least positive value of $x$ for which $\cos (\frac{x}{2} + \frac{π}{3}) = \frac{1}{\sqrt{2}}$ .
22M.2.AHL.TZ2.7a:
Show that a finite limit only exists for $k = \frac{π}{4}$ .
19N.1.AHL.TZ0.H_9c.ii:
By considering triangle ${\text{CBD}}$ , find another expression for ${\text{B}}{{\text{D}}^2}$ in terms of $r$ and $q$ .
17N.1.SL.TZ0.S_4a:
Show that ${\text{AC}} = 7{\text{ cm}}$ .
16N.2.AHL.TZ0.H_7a:
Use the cosine rule to find the two possible values for AC.
16N.1.AHL.TZ0.H_13a:
Find the value of $\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}$ .
19N.1.AHL.TZ0.H_9a:
Given that ${\text{cos}}\,75^\circ = q$ , show that ${\text{cos}}\,105^\circ = - q$ .
17M.1.SL.TZ1.S_3:
The following diagram shows triangle PQR.

Find PR.
19M.1.AHL.TZ1.H_4b:
Find the two possible values for the length of the third side.
17N.1.SL.TZ0.S_4b:
The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.

Find the exact perimeter of this shape.
18N.2.SL.TZ0.S_7b:
Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.
21N.2.SL.TZ0.3:
Consider a triangle $ABC$ , where $AC = 12, CB = 7$ and $B \hat{A} C = 25 °$ .

Find the smallest possible perimeter of triangle $ABC$ .

Hide related questions

Topic 3— Geometry and trigonometry » AHL 3.9—Reciprocal trig ratios and their pythagorean identities. Inverse circular functions

Topic 5 —Calculus » SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves

Topic 5 —Calculus » AHL 5.16—Integration by substitution, parts and repeated parts

Topic 3— Geometry and trigonometry

Topic 5 —Calculus