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Trigonometric substitution

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Integration by trigonometric substitution is a technique of integration that involves making a change of variables for example normally one would change from the variable x to theta.

Examples: Compute \int \sqrt{1-x^2} dx
For this problem it seems wise to let x=\sin{\theta},dx=\cos{\theta}d\theta
Now with this substitution we have: \int \sqrt{1-\sin^2{\theta}}\cos{\theta}d\theta
By the trigonometric identity \sin^2{x}+\cos^2{x}=1 we now have:
\sqrt{1-\sin^2{\theta}}=\cos{\theta}
Therefore:
\int \cos^2{\theta} d\theta

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