Integration: Definite Integrals 4

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Ben McGovern 2 years, 7 months ago

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Ben McGovern 4 years, 8 months ago

Created this as a copy of Integration: Definite Integrals 3.

Name	Status	Author	Last Modified
Differentiation: Using a Table of Derivatives 2	Ready to use	Ben McGovern	26/09/2022 12:10
Differentiation: Using a Table of Derivatives 3	Ready to use	Ben McGovern	26/09/2022 12:11
Differentiation: Using a Table of Derivatives 4	Ready to use	Ben McGovern	26/09/2022 12:11
Differentiation: Using a Table of Derivatives 5a	Ready to use	Ben McGovern	26/09/2022 12:11
Differentiation: Using a Table of Derivatives 5b	Ready to use	Ben McGovern	26/09/2022 12:12
Differentiation: Using a Table of Derivatives 5c	Ready to use	Ben McGovern	26/09/2022 12:12
Differentiation (Harder powers - negative and fractions) 2	Ready to use	Ben McGovern	04/05/2023 05:42
Differentiation: Stationary Points and their Nature 1	Ready to use	Ben McGovern	26/09/2022 11:56
Differentiation: Stationary Points and their Nature 2	Ready to use	Ben McGovern	26/09/2022 11:56
Integration: Using a Table of Integrals 2	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 3a	Ready to use	Ben McGovern	13/02/2023 10:37
Integration: Using a Table of Integrals 3b	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 4	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 5a	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 5b	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 5c	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 6	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 1a i	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 1b	Ready to use	Ben McGovern	23/02/2023 12:45
Integration: By Substitution 1c	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 2a	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 2b	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 2c	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 3a i	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 3c	Ready to use	Ben McGovern	27/09/2022 09:01
Integration (Negative and Fractional Powers) 2	Has some problems	Ben McGovern	27/09/2022 09:01
Integration (Negative and Fractional Powers) 3	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Definite Integrals 2	Ready to use	Ben McGovern	26/09/2022 12:18
Integration: By Substitution 1a ii (Negative and Fractional Powers)	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 3a ii (Negative and Fractional Powers)	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Definite Integrals 3	Ready to use	Ben McGovern	26/09/2022 12:18
Integration: Definite Integrals 4	Ready to use	Ben McGovern	26/09/2022 12:19
CB2 Differentiating polynomials 2	Ready to use	Mash Sheffield	17/04/2025 09:02
CD3 Integration - Substitution	Ready to use	Andrew Neate	17/04/2025 09:02
CB4 Definite integration	Ready to use	Andrew Neate	17/04/2025 09:02
CB6 Differentiating Trig 1	Ready to use	Andrew Neate	17/04/2025 09:02
CB7 Differentiating Trig 2	Ready to use	Andrew Neate	17/04/2025 09:02
CB8 Differentiating with Trig 3	Ready to use	Andrew Neate	17/04/2025 09:02
CB9 Differentiating with Exponentials	Ready to use	Andrew Neate	17/04/2025 09:02
CB10 Differentiation with logs	Ready to use	Andrew Neate	17/04/2025 09:02
CB3 - Indefinite integration - polynomials	Ready to use	Andrew Neate	17/04/2025 09:02
CB5 - Finding turning points	Ready to use	Andrew Neate	17/04/2025 09:02

There are 151 other versions that do you not have access to.

Question statement

Evaluate

$\int_\simplify{{n_1}pi/12}^\simplify{{n_2}pi/12}\simplify{{a}cos({b}x)}dx$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
b	integer	3
n_1	integer	4
n_2	integer	24
simp1	rational	1
simp2	rational	6
frac1	number	0
frac2	number	0
sol	number	0.00

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1. Gap Gap 0 Mathematical expression
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Advice

Integrating a function of the form

$f(x)=\cos(kx)$ has the integral

$\int_a^b \cos(kx) dx = \left[\frac{1}{k}\sin(kx)\right]_a^b,$

and

$\int_a^b kf(x) dx = k \int_a^b f(x) dx.$

Therefore,

$\begin{split} \simplify{defint({a}cos({b}x),x,{n_1}pi/12,{n_2}pi/12)} &\,= \simplify{{a}defint(cos({b}x),x,{n_1}pi/12,{n_2}pi/12)}, \\ &\,= \left[ \simplify{{a}/{b}sin({b}x)}\right]_\simplify{{n_1}pi/12}^\simplify{{n_2}pi/12}, \\ &\,= \left[\simplify[all,!trig,fractionNumbers]{{a/b}sin({simp2}pi)}\right]-\left[\simplify[all,!trig,fractionNumbers]{{a/b} sin({simp1}pi)}\right] \\&\,=\simplify{{sol}} \end{split}$

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This question is used in the following exams:

Integration: Definite Integrals by Ben McGovern in MASH Bath: Moodle quizzes and TS.
Integration - Definite Integrals by Yousuf Ibrahim in Yousuf's workspace.
All the good stuff by Chris Templet in Chris's workspace.
Week 14 e-quiz by Ruth Douglas in Ruth's workspace.
Summer Quiz 4 by Kate Vincent in Kate's workspace.
Sam's copy of Integration - Definite Integrals - Worksheet by Sam jemison in Sam's workspace.
Subject Knowledge Homework 2 by Ciaran Cassidy in A Level Subject Knowledge Homework.
Calc Practice by S Russell in S's workspace.

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Ben McGovern 2 years, 7 months ago

Ben McGovern 4 years ago

Ben McGovern 4 years, 8 months ago

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Integration: Definite Integrals 4

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 2 years, 7 months ago

Ben McGovern 4 years ago

Ben McGovern 4 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`