Trigonometric identities in integration

This article continues from a previous article. In this article we shall look at how to use trigonometric identities in integration. Below is a list of trigonometric identities that have been integrated. You should be familiar with the list to integrate trigonometric functions.
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You can take advantage of trigonometric identities by replacing trigonometric functions with trigonometric functions you know how to integrate. We shall look at some examples below;

Example

Find;
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You cannot integrate tan²x, to make it easier you replace it with another trigonometric function you know how to integrate. So you think of a trigonometric identity that involves tanx. Remember that;
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…that must mean that;
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We can integrate sec²x so replace tan²x in the integration;
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…we know that;
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…so we get;
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Example

Find;
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Again you cannot integrate sin²x, so you’ll need to find a trigonometric identity that you can integrate. Remember;
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We can rearrange the identity to get;
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…therefore;
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Example

Find;
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He we can use the identity;
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…therefore;
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Example

Find
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We will need to simplify the expression here;
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Remember tan²x = sec²x – 1, therefore;
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We now have a standard integral so we can integrate;
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Example

Find;
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Here we will need to use the following identity. We know that;
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…that is…
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…and…
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If we add both expressions we get;
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So we have;
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