Integration

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

Level 1Level 2Level 3Level 4Level 5Level 6Exam-StyleDescriptionHelpDifferentiation

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.

 $$\int 3x^2 \; \text{dx}$$ = $$\int 6x^5 \; \text{dx}$$ = $$\int 10x^4 \; \text{dx}$$ = $$\int 7 \; \text{dx}$$ = $$\int 14x - 25 \; \text{dx}$$ = $$\int 3x^2 + 6x^5 \; \text{dx}$$ = $$\int 20x^4 - 8x^3 \; \text{dx}$$ = $$\int 12x^3 + 15x^2 - 6x \; \text{dx}$$ = $$\int 99x^2-22x+17 \; \text{dx}$$ =
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This is Integration level 1. You can also try:
Level 2 Level 3 Level 4 Level 5 Level 6

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

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

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

Tutorial

The video above is from the wonderful HEGARTYMATHS

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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