Search any question & find its solution
Question:
Answered & Verified by Expert
By using the properties of definite integrals, evaluate the integrals
$\int_0^1 x(1-x)^n d x=\mathrm{I}$ (say)
$\int_0^1 x(1-x)^n d x=\mathrm{I}$ (say)
Solution:
1591 Upvotes
Verified Answer
Let $\mathrm{I}=\int_0^1(1-x)[1-(1-x)]^n d x=\int_0^1\left(x^n-x^{n-1}\right) d x$
$=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_0^1=\left(\frac{1}{(n+1)(n+2)}\right)$
$=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_0^1=\left(\frac{1}{(n+1)(n+2)}\right)$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.