मराठी

By using the properties of the definite integral, evaluate the integral: ∫0π2 sinxsinx+cosxdx

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प्रश्न

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2)  sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx` 

बेरीज
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उत्तर

Let `I = int_0^(pi/2) sqrtsinx/(sqrt sinx + sqrt cos x)  dx`     ...(i)

Replace x to `(pi/2 - x)` in (i)

`[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`

`I = int_0^(pi/2) (sqrt sin (pi/2 - x))/ (sqrt sin (pi/2 - x) + sqrt cos (pi/2 - x))  dx`

`I = int_0^(pi/2) sqrtcosx/(sqrtcos x + sqrt sin x)  dx`       ...(ii)

Adding (i) and (ii), we get

`2I = int_0^(pi/2) [sqrt sinx/ (sqrt sinx + sqrt cos x) + sqrt cos x/(sqrt cos x + sqrt sinx)]  dx` 

`= int_0^(pi/2) (sqrt cos x + sqrt sin x)/(sqrt cosx + sqrt sin x)`

`= int_0^(pi/2) dx = [x]_0^(pi/2)`

`= pi/2 - 0`

`= pi/2`

⇒ `I = pi/4`

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पाठ 7: Integrals - Exercise 7.11 [पृष्ठ ३४७]

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एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12
पाठ 7 Integrals
Exercise 7.11 | Q 2 | पृष्ठ ३४७

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