{"id":7751,"date":"2021-11-05T23:58:49","date_gmt":"2021-11-05T18:28:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7751"},"modified":"2021-11-06T01:35:51","modified_gmt":"2021-11-05T20:05:51","slug":"what-is-the-integration-of-x-sin-inverse-x-dx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-integration-of-x-sin-inverse-x-dx\/","title":{"rendered":"What is the integration of x sin inverse x dx ?"},"content":{"rendered":"
We have, I = \\(\\int\\)\u00a0 \\(x sin^{-1} x\\) dx<\/p>\n
By using integration by parts formula,<\/p>\n
I = \\(sin^{-1} x\\) \\(x^2\\over 2\\) – \\(\\int\\) \\(1\\over \\sqrt{1 – x^2}\\) \\(\\times\\) \\(x^2\\over 2\\) dx<\/p>\n
I =\u00a0 \\(x^2\\over 2\\) \\(sin^{-1} x\\) + \\(1\\over 2\\) \\(\\int\\) \\(-x^2\\over \\sqrt{1 – x^2}\\) dx<\/p>\n
= \\(x^2\\over 2\\) \\(sin^{-1} x\\) + \\(1\\over 2\\) \\(\\int\\) \\(1 – x^2 – 1\\over \\sqrt{1 – x^2}\\)\u00a0 dx<\/p>\n
= \\(x^2\\over 2\\) \\(sin^{-1} x\\) + \\(1\\over 2\\) { \\(\\int\\) \\(1 – x^2\\over \\sqrt{1 – x^2}\\) – \\(\\int\\) \\(1\\over \\sqrt{1 -x^2}\\) } dx<\/p>\n
\\(\\implies\\) I = \\(x^2\\over 2\\) \\(sin^{-1} x\\) + \\(1\\over 2\\) { \\(\\int\\) \\(\\sqrt{1 – x^2}\\) – \\(\\int\\) \\(1\\over \\sqrt{1 -x^2}\\) } dx<\/p>\n
By using integration formula of \\(\\sqrt{a^2 – x^2}\\),<\/p>\n
\\(\\implies\\) I = \\(x^2\\over 2\\) \\(sin^{-1} x\\) + \\(1\\over 2\\) [{ \\(1\\over 2\\) \\(x\\sqrt{1 – x^2}\\) – \\(1\\over 2\\) \\(sin^{-1} x\\) } – \\(sin^{-1} x\\) ] + C<\/p>\n
\\(\\implies\\) I = \\(x^2\\over 2\\) \\(sin^{-1} x\\) +\u00a0 \\(1\\over 4\\) \\(x\\sqrt{1 – x^2}\\) – \\(1\\over 4\\) \\(sin^{-1} x\\) + C<\/p>\n
What is the integration of sin inverse root x ?<\/a><\/p>\n What is the integration of sin inverse x whole square ?<\/a><\/p>\n What is integration of sin inverse cos x ?<\/a><\/p>\n What is the integration of tan inverse root x ?<\/a><\/p>\n