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Q)

Evaluate the following problems using properties of integration: $\int\limits_{0}^{1} x(1-x)^{10}dx$

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A)
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  • $\int \limits_0^a f(x) dx=\int \limits_0^a f(a-x) dx$
$\int\limits_{0}^{1} x(1-x)^{10}dx$
Step 1:
$I=\int\limits_{0}^{1} x(1-x)^{10}dx$
Step 2:
$\qquad=\int\limits_{0}^{1} x(1-x)(1-(1-x))^{10}dx$
$\qquad=\int\limits_{0}^{1} (1-x)x^{10}dx$
Step 3:
$\qquad=\int\limits_{0}^{1} (x^{10}-x^{11})dx$
$\qquad=\large\frac{x^4}{11}-\frac{x^{12}}{12} \bigg]_0^1$
$\qquad= \large\frac{1}{11}-\frac{1}{12}$
$\qquad=\large\frac{1}{132}$
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