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Integration

Exercises on indefinite and definite integration of basic algebraic and trigonometric functions.

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This is level 8 ?  Your answer should be a number (rounded to three significant figures if not exact) or a fraction in its lowest terms.

\(\int _1^2 2x(x^2+3)^5 \; \text{dx} \)

= Correct Wrong

\(\int _0^{\sqrt{3}} x\sqrt{1+x^2} \; \text{dx} \)

= Correct Wrong

\(\int _1^3 4x(x^2-2)^4 \; \text{dx} \)

= Correct Wrong

\(\int _2^3 \sqrt{x^3+3x}(3x^2+3) \; \text{dx} \)

= Correct Wrong

\(\int _0^1 5x \cos{x^2} \; \text{dx} \)

= Correct Wrong

\(\int _{0.25}^{0.75} \dfrac{1}{(5-4x)^3} \; \text{dx} \)

= Correct Wrong

\(\int _0^{\frac{\pi}{4}} \sec^2{x}\tan^3{x} \; \text{dx} \)

= Correct Wrong

\(\int _1^2 x^2 e^{x^3+1} \; \text{dx} \)

= Correct Wrong

\(\int _{\frac12}^{\frac{\sqrt{3}}{2}} \dfrac{4}{\sqrt{1-x^2}} \; \text{dx} \)

= Correct Wrong

For the last question use the substitution \(x = \sin{\theta}, \quad \text{where } -\frac{\pi}{2} \lt \theta \lt \frac{\pi}{2} \)

Check

This is Integration level 8. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 9 Differentiation

Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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Whose Idea Was This?

Did you enjoy doing this 'Integration' activity? Are you curious about who originally came up with this idea in Maths? Discover more about one of the mathematicians who is associated with this concept.

Featured Activity

Remainder Race

Remainder Race

A brilliant game involving chance and choice requiring an ability to calculate the remainder when a two digit number is divided by a single digit number. There are one and two player versions and the rules are inspired by the Royal Game of Ur.

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Go Maths

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Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.

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Description of Levels

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Level 1 - Indefinite integration of basic polynomials with integer coefficient solutions

Level 2 - Indefinite integration of basic polynomials with integer and fraction coefficient solutions

Level 3 - Definite integration of basic polynomials

Level 4 - Integration of expressions containing fractional indices

Level 5 - Integration of basic trigonometric, exponential and reciprocal functions

Level 6 - Integration of the composites of basic functions with the linear function ax + b

Level 7 - Integration with the help of partial fractions

Level 8 - Integration by substitution

Level 9 - Integration by parts

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

Differentiation - A multi-level set of exercises providing practice differentiating expressions

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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In General

$$\int ax^n \text{dx} = \frac{ax^{n+1}}{n+1}+c \quad \text{ for all } n \neq -1$$

Tutorial

Mathematical Notation

Use the ^ key to type in a power or index then the right arrow or tab key to end the power.

For example: Type 3x^2 to get 3x2.


Use the forward slash / to type a fraction then the right arrow or tab key to end the fraction.

For example: Type 1/2 to get ½.

Fractions should be given in their lowest terms.

Answers to definite integral questions should be given as exact fractions or to three significant figures if the decimal answer does not terminate.


Special Functions

$$\int e^x \; \text{dx} = e^x + c$$ $$\int \frac1x \; \text{dx} = \ln x + c$$ $$\int \cos x \; \text{dx} = \sin x + c$$ $$\int \sin x \; \text{dx} = -\cos x + c$$

Composite Functions

$$\int e^{ax+b} \; \text{dx} = \frac1a e^{ax+b} + c$$ $$\int (ax+b)^n \; \text{dx} = \frac1a \frac{(ax+b)^{n+1}}{n+1} + c \text{,} \quad (n \neq -1)$$ $$\int \frac{1}{ax+b}\; \text{dx} = \frac1a \ln (ax+b)+ c \text{,} \quad (ax+b \gt 0)$$ $$\int \cos (ax+b) \; \text{dx} = \frac1a \sin (ax+b) + c$$ $$\int \sin (ax+b) \; \text{dx} = - \frac1a \cos (ax+b) + c$$

The following identities may also prove useful:

$$\sin^2x = \frac{1}{2} - \frac{1}{2} \cos 2x \text{ and } \cos^2x = \frac{1}{2} + \frac{1}{2} \cos 2x$$

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Typing Mathematical Notation

These exercises use MathQuill, a web formula editor designed to make typing Maths easy and beautiful. Watch the animation below to see how common mathematical notation can be created using your keyboard.

MathQuill Animation

Integration Flowchart

Integration Flowchart

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