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

Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Exam-Style Description 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.

\(\int 3x^2 \; \text{dx} \)

= Correct Wrong

\(\int 6x^5 \; \text{dx} \)

= Correct Wrong

\(\int 10x^4 \; \text{dx} \)

= Correct Wrong

\(\int 7 \; \text{dx} \)

= Correct Wrong

\(\int 14x - 25 \; \text{dx} \)

= Correct Wrong

\(\int 3x^2 + 6x^5 \; \text{dx} \)

= Correct Wrong

\(\int 20x^4 - 8x^3 \; \text{dx} \)

= Correct Wrong

\(\int 12x^3 + 15x^2 - 6x \; \text{dx} \)

= Correct Wrong

\(\int 99x^2-22x+17 \; \text{dx} \)

= Correct Wrong


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


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Featured Activity



This activity comprises of five challenges each one requiring you to type in one more fraction than the previous challenge and each starting with a different fraction. The starting fractions are one tenth, one quarter, one third, one half and four sevenths.


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Description of Levels



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

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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In General

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


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:

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

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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