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प्रश्न
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
उत्तर
Let I = `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x).dx` ...(i)
= `int_2^5 (sqrt(2 + 5 - x))/(sqrt(2 + 5 - x) + sqrt(7 - (2 + 5 - x))).dx` ...`[∵ int_a^b f(x)dx = int_a^b f(a + b - x)dx]`
∴ I = `int_2^5 (sqrt(7 - x))/(sqrt(7 - x) + sqrt(x)).dx` ...(ii)
Adding equations (i) and (ii), we get
2I = `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x).dx + int_2^5 sqrt(7 - x)/(sqrt(7 - x) + sqrt(x)).dx`
= `int_2^5 (sqrt(x) + sqrt(7 - x))/(sqrt(x) + sqrt(7 - x)).dx`
= `int_2^5 1.dx`
= `[x]_2^5`
∴ 2I = 5 – 2
2I = 3
I = `3/2`
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