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  "body": "\n## Summary\n\nTrigonometric substitution is a [[substitution|0D37QFPEZE]] method for integrals that contain $\\sqrt{a^2 - x^2}$, $\\sqrt{a^2 + x^2}$, or $\\sqrt{x^2 - a^2}$. We set $x$ equal to $a \\sin \\theta$, $a \\tan \\theta$, or $a \\sec \\theta$. The [[Pythagorean identity|03H7WZ4T3K]] then removes the square root.\n\n## Remarks\n\nWith $x = a \\sin \\theta$ for $\\theta \\in [-\\pi/2, \\pi/2]$, we get $\\sqrt{a^2 - x^2} = a \\cos \\theta$, for $a \u003e 0$.\nFor example, this substitution gives $\\int_{-1}^{1} \\sqrt{1 - x^2} \\, dx = \\pi/2$, the area of a half disc.\nHyperbolic functions can replace the trigonometric functions in the second and the third case.\n\n## Explanations\n\n### L1 human\n\nA square root such as $\\sqrt{1 - x^2}$ is the side of a right triangle.\nWriting $x$ as the sine of an angle turns the root into a cosine.\n",
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