The correct option is A a^n+2/(n+1)(n+2)∫_0^a x(a x)^n dx=∫_0^a(a x)x^n dx= [ ax^n+1/n+1 x^n+1/n+2 ]_0^a= [ 1/n+1 1/n+2 ]=a^n+2/(n+1)(n+2) ...

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Inequalities of Integrals

∫0a xa-xn dx=

Question

$\int_{0}^{a} x (a - x)^{n} d x =$

\frac{a^{n + 2}}{(n + 1) (n + 2)}

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\frac{2 a^{n + 2}}{(n + 1) (n + 2)}

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\frac{a^{n + 2}}{(n - 1) (n + 2)}

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\frac{a^{n + 2}}{(n + 1) (n - 2)}

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Solution

The correct option is A $\frac{a^{n + 2}}{(n + 1) (n + 2)}$
$\int_{0}^{a} x (a - x)^{n} d x = \int_{0}^{a} (a - x) x^{n} d x = {[\frac{a x^{n + 1}}{n + 1} - \frac{x^{n + 1}}{n + 2}]}_{0}^{a} = [\frac{1}{n + 1} - \frac{1}{n + 2}] = \frac{a^{n + 2}}{(n + 1) (n + 2)}$

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Similar questions

\int_{0}^{a} x^{2} (a - x)^{n} d x = \frac{2 a^{n + 3}}{(n + 1) (n + 2) (n + 3)}

The number of terms in the expansion of (a + b + c)ⁿ are:

\sum_{k = 1}^{n - 1} (n - k) c o s \frac{2 k π}{n} = - \frac{n}{2}

n \geq 3

(1 + x)^{n} = n \sum r = 0^{n} C_{r} x^{n},

\frac{C_{0}}{1 \cdot 2} 2^{2} + \frac{C_{1}}{2 \cdot 3} 2^{3} + \frac{C_{2}}{3 \cdot 4} 2^{4} + \dots + \frac{C_{n}}{(n + 1) (n + 2)} 2^{n + 2}

1 \times n + 2 (n - 1) + 3 (n - 2) + \dots \dots + (n - 1) \times 2 + n \times 1

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