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दिखाएँ कि int(0)^(a) f(x) g (x) dx = 2...

दिखाएँ कि ` int_(0)^(a) f(x) g (x) dx = 2 int _(0)^(a) f(x) dx, ` यदि f तथा g निम्न प्रकार परिभाषित है ।
`f(x) = f(a-x) " तथा " g (x) + g (a-x) =4.0`

लिखित उत्तर

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दिया है `g(x) +g(a-x) = 4 " " ` …(1)
तथा ` f(x) = f(a-x)" "` …(2)
माना कि ` I = int_(0)^(a) f(x) g (x) dx `
तो , ` I = int _(0)^(a) f(a-x) g ( a-x) dx " " [ :' int_(0)^(a) f(x) dx = int_(0)^(a) f(a- x) dx ] `
` = int _(0)^(a) f(x) [ 4 - g(x)] dx " "` [ (1) और (2) से]
` = 4 int _(0)^(a) f(x) dx - int _(0)^(a) f(x) g(x) dx `
या ` I = 4 int _(0)^(a) f(x) dx -I " या " , 2I = 4 int _(0)^(a) f(x) dx`
` :. I = 2 int _(0)^(a) f(x) dx`
या , ` int _(0)^(a) f(x) g(x) dx = 2 int_(0)^(a) f(x) dx `
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KC SINHA-निश्चित समाकलों के गुणधर्म -OBJECTIVE QUESTIONS
  1. दिखाएँ कि int(0)^(a) f(x) g (x) dx = 2 int (0)^(a) f(x) dx, यदि ...

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  2. int (0)^(pi) cos^(2m+1)" x " dx , where m is a positive integar is eq...

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  3. The value of int (0)^(pi//2) (sqrt(cot x ))/(sqrt(tan x) + sqrt(cot ...

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  4. int (a)^(2a) f(x) dx = 2 int (0)^(a) f(x) dx if

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  5. यदिg(x) = int (0)^(x) cos4tdt फिरg(x + pi) बराबर होगा

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  6. Let f(x) = x - [x] , for every real number x, where [ x] is the integr...

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  7. int (-pi)^(pi) (cos^(2)x)/(1+a^(x)) dx, a gt 0

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  8. Let f, R to R and g : R to R be continous functions . Then the value...

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  9. The value of int (-1)^(2) (|x|)/x dx is

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  10. int (0)^(pi) sqrt((1+cos 2 x)/2) dx is

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  11. For any integar n, the integral int (0)^(pi) e^(sin ^(2)x) . cos^(3)...

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  12. ज्ञात करे । int (pi//4)^(3pi//4) (phi)/(1+sin phi) d phi =

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  13. निम्नलिखित समाकलन का मान निकाले int (0)^(pi//2) (dx)/(1+tan^(3) x )

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  14. int (-1//2)^(1//2) cos x log ((1+x)/(1-x)) dx =

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  15. The value of the integral int (0)^(pi//2) | cos x - sin x | dx is

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  16. int (0)^(2) | x-1| dx is equal to

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  17. The value of the integral int (0)^(pi//2) (sqrt(cos x))/(sqrt(cosx)+ ...

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  18. The value of int (-pi//2)^(pi//2) sqrt((1-cos 2theta )/2 ) d theta ...

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  19. int (-pi//2)^(pi//2) sin^(2) x cos^(2) x ( sin x + cos x ) dx =

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  20. int (-1)^(1) | 1- x| dx =

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  21. int (0)^(pi//2) 1/(1+tan theta ) d theta =

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