Integration
Integration is the reverse of differentiation.
After differentiating a function, integration can be used to get the original function back. This concept is called the Fundamental Theorem of calculus. Knowledge of the principles of differentiation is vital here since integration is the opposite of differentiation.
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You must know that differentiation is the process of finding a derivative (or finding the gradient or gradient function of a curve). Integration can be used in many other areas such as finding areas and volumes.
We shall explore different integration operations in this topic. Below is a list of articles in this topic. Click on any of the list to read through or click next to start. At the bottom of the page is a quick summary of the basic rules of integration or integrals that you should be familiar with;
- Integration basics
- Area under a curve
- Area under a curve with negative values
- Area between a straight line and a curve
- Integration using the reverse chain rule
- Trigonometric identities in integration
- Partial fractions in integration
- Using standard patterns to integrate expressions
- Integration by substitution
- Integration by parts
- Numerical integration
- Integration to find areas and volumes
- Solving differential equations using integration
- Differential equations in context
Introduction
Below is a summary of rules of integration you should be familiar with;
If;
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…we can express this as;
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We can also take advantage of the reverse of the chain rule to obtain the following rules;
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Trigonometric identities can be helpful in integration. You can change certain expressions into expressions you know to integrate. For example it is easier to integrate sin²x or cos²x by using the formulae cos2x, so;
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Using partial fractions to integrate expressions of the following form is useful. The expression is first converted into a partial fraction;
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The general patterns that you’ll learn in this topic;
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A process known as integration by parts can be used to integrate expressions;
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The definite integral;
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The area beneath the curve with equation y = f(x) and between the lines x=a and x=b is;
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The area between a line (equation y1) a curve (equation y2) is given by;
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The area of region between y=f(x), the x-axis and x=a and x=b is given by;
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The volume of revolution formed by rotating y about the x-axis between x=a and x=b is given by;
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You can use the “separating the variables” method in integration, where … when;
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…then we can write;
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please note that there is a mistake while integrating 2x^2 dx the result should be 2x^3 / 3 +c.