# Integration

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

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This is level 2 ?  Use the ^ key to type in a power or index and use the forward slash / to type a fraction. Press the right arrow key to end the power or fraction. Click the Help tab above for more.

Each of your answers should end with +c for the constant of integration.

 $$\int 2x^2 \; \text{dx}$$ = $$\int 5x^3 \; \text{dx}$$ = $$\int 2x^5 \; \text{dx}$$ = $$\int 2x^3-5x \; \text{dx}$$ = $$\int 9x^4+x^2 \; \text{dx}$$ = $$\int 7x-35 \; \text{dx}$$ = $$\int \frac{5}{x^2} \; \text{dx}$$ = $$\int \frac{7}{x^3} \; \text{dx}$$ = $$\int -\frac{1}{x^4} \; \text{dx}$$ =
Check

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

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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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#### Hi-Low Predict

A version of the Play Your Cards Right TV programme. Calculate the probabilities of cards being higher or lower than the one shown. a fun way to practise applying probability and using fractions.

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## Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

## 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

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## In General

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

## 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:

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

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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.

Integration Flowchart

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