Evaluate : _0^1 x(1-x)^n d x; ( where n N) - Vidyadip

Question

Evaluate : $\int_0^1 x(1-x)^n d x$; $($ where $n \in N)$

✓

Answer

$\int_0^1 x(1-x)^n d x$
$\begin{array}{c}=\int_0^1(1-x)\{1-(1-x)\}^n d x,\left(a s, \int_0^a f(x) d x=\int_0^a f(a-x) d x\right) \\ =\int_0^1 x^n(1-x) d x \\ =\int_0^1 x^n d x-\int_0^1 x^{n+1} d x \\ =\frac{1}{n+1}\left[x^{n+1}\right]_0^1-\frac{1}{n+2}\left[x^{n+2}\right]_0^1 \\ =\frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{(n+1)(n+2)}\end{array}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from MODEL PAPER 2025→MODEL PAPER 2025 — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If the percentage error in the radius of a sphere is $\alpha,$ find the percentage error in its volume.

Find the area of the triangle with vertices at the points: $(3, 8), (-4, 2)$ and $(5, -1)$

Find the distance of the plane 2x- 3y + 4z - 6 = 0 from the origin.

Of the students in a college, it is known that $60\%$ reside in a hostel and $40\%$ do not reside in hostel. Previous year results report that $30\%$ of students residing in hostel attain A grade and $20\%$ of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteler?

If function $f(x)=\left\{\begin{array}{cc}\frac{x^2-2 x-3}{x+1}, & : x \neq 1 \\ \lambda & : x=-1\end{array}\right.$ is continuous at $x = - 1$ then find value of $\lambda$.

Show that $f(x) = x^9 + 4x^7 + 11$ is an increasing function for all $\text{x}\in\text{R}.$

Prove that $\text{f(x)}=\sin\text{x}+\sqrt{3}\cos\text{x}$ has maximum value at $\text{x}=\frac{\pi}{6}.$

Solve the following equation for x:
$\tan^{-1}\frac{\pi}{2}+\tan^{-1}\frac{\pi}{3}=\frac{\pi}{4},0<\text{x}<\sqrt6$

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).

Evaluate the following integrals:
$\int\tan^\frac{3}{2}\text{x}\sec^2\text{x dx}$