# Integration

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

##### Menu  Level 1L2L3L4L5L6L7L8L9  Exam    Help  Differentiation

This is level 1 ?  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.

 $$\int 3x^2 \; \text{dx}$$ = $$\int 6x^5 \; \text{dx}$$ = $$\int 10x^4 \; \text{dx}$$ = $$\int 7 \; \text{dx}$$ = $$\int 14x - 25 \; \text{dx}$$ = $$\int 3x^2 + 6x^5 \; \text{dx}$$ = $$\int 20x^4 - 8x^3 \; \text{dx}$$ = $$\int 12x^3 + 15x^2 - 6x \; \text{dx}$$ = $$\int 99x^2-22x+17 \; \text{dx}$$ =
Check

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

## Instructions

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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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