Show that sec 2 x + sec x tan x = 1 1 - sin x .

To prove: sec 2 x + sec x tan x = 1 1 - sin x Consider, RHS, 1 1 - sin x = 1 1 - sin x × 1 + sin x 1 + sin x = 1 + sin x 1 - sin 2 x = 1 + sin x cos 2 x = 1 cos 2 x + sin x cos 2 x = sec 2 x + sec x tan x = LHS Hence, proved.

If y = sec x , then show that d y d x = sec x tan x .

Given that, y = sec x ⇒ y = 1 cos x and to prove: d y d x = sec x tan x Differentiate y with respect to x , then d y d x = cos x d d x 1 - d d x cos x cos x 2 [Using Quotient Rule in Differentiation] ⇒ d y d x = sin x cos 2 x ⇒ d y d x = sin x cos x × 1 cos x ⇒ d y d x = sec x tan x Hence, proved.

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5D

Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Integration, Exercise 5: EXERCISE 5D

Author:Sue Pemberton, Julianne Hughes & Julian Gilbey

Sue Pemberton Mathematics Solutions for Exercise - Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Integration, Exercise 5: EXERCISE 5D

Attempt the free practice questions on Chapter 5: Integration, Exercise 5: EXERCISE 5D with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book solutions are prepared by Experienced Embibe Experts.

Questions from Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Integration, Exercise 5: EXERCISE 5D with Hints & Solutions

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5D>Q 7

Show that $\sin 3 x = 3 \sin x - 4 \sin^{3} x$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5D>Q 7

Find the value of $\int_{0}^{\frac{1}{3} π} 2 \sin^{3} x d x$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5D>Q 8

Show that $\cos 3 x = 4 \cos^{3} x - 3 \cos x$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5D>Q 8

If the value of $\int_{0}^{\frac{1}{6} π} (4 \cos^{3} x + 2 \cos x) d x$ is $\frac{k}{6}$ , find the value of $k$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5D>Q 9

Question Image

The diagram shows part of the curve $y = \cos x + 2 \sin x$ . Find the exact volume of the solid formed when the shaded region is rotated through $360 °$ about the $x -$ axis.