Definite Integration
Grade 12
Question:
<p>Evaluate \(\displaystyle\int_0^1(1-x)^n\,dx\) [JEE Main 2015]</p>
Step-by-Step Solution
Key Concept: Let t = 1-x: \int_0^1(1-x)ⁿdx = \int_0^1 tⁿ dt = 1/(n+1).
<div class='solution'>
<p>Let $t=1-x\Rightarrow dt=-dx$. Limits: $x=0\to t=1$; $x=1\to t=0$.</p>
<p>$$\int_0^1(1-x)^n dx=\int_1^0 t^n(-dt)=\int_0^1 t^n dt=\frac{1}{n+1}$$</p>
Correct Answer: A