# Integration

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

##### Menu  L1L2L3L4L5L6L7Level 8L9  Exam    Help  Differentiation

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.

 $$\int _1^2 2x(x^2+3)^5 \; \text{dx}$$ = $$\int _0^{\sqrt{3}} x\sqrt{1+x^2} \; \text{dx}$$ = $$\int _1^3 4x(x^2-2)^4 \; \text{dx}$$ = $$\int _2^3 \sqrt{x^3+3x}(3x^2+3) \; \text{dx}$$ = $$\int _0^1 5x \cos{x^2} \; \text{dx}$$ = $$\int _{0.25}^{0.75} \dfrac{1}{(5-4x)^3} \; \text{dx}$$ = $$\int _0^{\frac{\pi}{4}} \sec^2{x}\tan^3{x} \; \text{dx}$$ = $$\int _1^2 x^2 e^{x^3+1} \; \text{dx}$$ = $$\int _{\frac12}^{\frac{\sqrt{3}}{2}} \dfrac{4}{\sqrt{1-x^2}} \; \text{dx}$$ =

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

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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

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#### Nine Digits

Arrange the given digits one to nine to make three numbers such that two of them add up to the third. This is a great puzzle for practicing standard pen and paper methods of three digit number addition and subtraction.

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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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