Integration

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

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Use partial fractions to find these integrals. The structure of the required answer is given.

 $$\int \dfrac{1}{(x+2)(x+1)} \; \text{dx}$$ = $$\int \dfrac{2}{x(x-2)} \; \text{dx}$$ = $$\int \dfrac{3x-4}{x^2-3x+2} \; \text{dx}$$ = $$\int \dfrac{4x+1}{3x^2+5x-2} \; \text{dx}$$ = $$\int \dfrac{2x+3}{x^2+x-2} \; \text{dx}$$ = $$\int \dfrac{6x^2+22x+18}{(x+1)(x+2)(x+3)} \; \text{dx}$$ = $$\int \dfrac{x}{x+1} \; \text{dx}$$ = $$\int \dfrac{2x^2-4}{x^2-1} \; \text{dx}$$ = $$\int \dfrac{2x^2+4x+8}{(2x-1)(x^2-1)} \; \text{dx}$$ =
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This is Integration level 7. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 8 Level 9 Differentiation

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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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