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प्रश्न
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
मूल्यांकन
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उत्तर
Let I = `int_0^1 x(1 - x)^5 *dx`
= `int_0^1 (1 - x)[1 - (1 - x)]^5*dx ...[because int_0^"a" f(x)*dx = int_0^"a" f("a" - x)*dx]`
= `int_0^1 (1 - x)x^5*dx`
= `int_0^1(x^5 - x^6)*dx`
= `int_0^1 x^5*dx - int_0^1 x^6*dx`
= `[(x^6)/6]_0^1 - [(x^7)/7]_0^1`
= `(1)/(6) (1^6 - 0) - (1)/(7) (1^7 - 0)`
= `(1)/(6) - (1)/(7)`
∴ I = `(1)/(42)`
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Notes
The textbook answer is incorrect. Answer given in the textbook is `1/4^2`. However, as per our calculation, it is `1/42`.
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