Evaluate the following: d∫12dx(x-1)(2-x) - Mathematics

प्रश्न

Evaluate the following:

`int_1^2 ("d"x)/sqrt((x - 1)(2 - x))`

योग

उत्तर

Let I = `int_1^2 ("d"x)/sqrt((x - 1)(2 - x))`

= `int_1^2 ("d"x)/sqrt(2x - x^2 - 2 + x)`

= `int_1^2 ("d"x)/sqrt(-x^2 + 3x - 2)`

= `int_1^2 ("d"x)/sqrt(-(x^2 - 3x + 2)`

= `int_1^2 ("d"x)/sqrt(-(x^2 - 3x + 9/4 - 9/4 + 2))` .....[Making perfect square]

= `int_1^2 ("d"x)/sqrt(-[(x - 3/2)^2 - 1/4])`

= `int_1^2 ("dx)/sqrt(1/4 - (x - 3/2)^2)`

= `int_1^2 ("d"x)/sqrt((1/2)^2 - (x - 3/2)^2)`

= `[sin^-1 ((x - 3/2)/(1/2))]_1^2`

= `[sin^-1 ((2x - 3)/1)]_1^2`

= `sin^-1 (4 - 3) - sin^-1 (2 - 3)`

= `sin^-1 (1) - sin^-1 (-1)`

= `sin^-1 (1) + sin^-1 (1)`

= `2 sin^-1 (1)`

= `2 xx pi/2`

= `pi`

Hence, I = `pi`.

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Evaluation of Simple Integrals of the Following Types and Problems

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 31 Q 30 Q 32

अध्याय 7: Integrals - Exercise [पृष्ठ १६५]

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एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12

अध्याय 7 Integrals
Exercise | Q 31 | पृष्ठ १६५

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