What is ∫₀¹x(1-x)^{9}dx equal to?

Correct Answer is : 1/1101/1101/110

Correct Answer is : 1/1101/1101/110

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UPSC NDA I 2018 Mathematics Solved Paper

Question : 78 of 120

Marks: +1, -0

What is

∫_{0}^{1}x(1-x)^{9}dx

equal to?

Solution:

Concept:

Use the property of definite integrals:

\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx

Explanation:

Let

f(x) = x(1-x)^9

Apply the property with

a=0

and

b=1

\int_{0}^{1} x(1-x)^9 dx = \int_{0}^{1} (1-x) \cdot (1-(1-x))^9 dx

Simplify the inner term:

1-(1-x) = x

So the integral becomes

\int_{0}^{1} (1-x) \cdot x^9 dx

Expand the integrand:

\int_{0}^{1} (x^9 - x^{10}) dx

Integrate term by term:

\left[ \frac{x^{10}}{10} - \frac{x^{11}}{11} \right]_{0}^{1}

Evaluate at the limits:

\left( \frac{1^{10}}{10} - \frac{1^{11}}{11} \right) - (0) = \frac{1}{10} - \frac{1}{11}

Simplify:

\frac{11-10}{110} = \frac{1}{110}

Answer:

\boxed{\frac{1}{110}}

(Option A).

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