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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\sqrt{\text{x}-\text{x}^2}\text{dx}$

Find a unit vector perpendicular to both the vectors $4\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $-2\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}.}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_{\frac{-\pi}{2}}\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)\text{dx}$

Using vectors, prove that in a $\Delta$ ABC,
$\frac{\text{a}}{\sin\text{A}}=\frac{\text{b}}{\sin\text{B}}=\frac{\text{c}}{\sin\text{C}}$
Where a, b and c are lengths of the sides opposite, respectively, to the angles A, B and C of $\Delta$ ABC.

Find the vector from the origin O to the centroid of the triangle whose vertices are (1, -1, 2), (2, 1, 3) and (-1, 2, -1).

If $\text{y}=\sin(\log\text{x})$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

verify that $\text{y}=\text{cx}+2\text{c}^2$ is a solution of the differential equation $2\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2-\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

Suppose $f_1$ and $f_2$ are non-zero one-one functions from R to R. Is $\frac{\text{f}_1}{\text{f}_2}$ necessarily one-one? Justify your answer. Here, $\frac{\text{f}_1}{\text{f}_2}:\text{R}\rightarrow\ \text{R}$ is given by $\Big(\frac{\text{f}_1}{\text{f}_2}\Big)(\text{x})=\frac{\text{f}_1(\text{x})}{\text{f}_2(\text{x})}$ for all $\text{x}\in\text{R}.$

Prove the following results:
$\frac{9\pi}{4}-\frac{9}{4}\sin^{-1}\frac{1}{3}=\frac{9}{4}\sin^{-1}\frac{2\sqrt2}{3}$

Show that f : R → R, given by f(x) = x - [x], is neither one-one nor onto.