By using the properties of definite integral, evaluate the integr… - Vidyadip

Question

By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{1}_{0}\text{x}(1-\text{x})^{\text{n}}\ \text{dx}$

✓

Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{1}\text{x}(1-\text{x})^{\text{n}}\ \text{dx}=\int\limits_{0}^{1}(1-\text{x)}\left\{1-(1-\text{x)}^{\text{n}}\right\}\text{dx}\ \bigg[\because\int\limits_{0}^{\text{a}}\text{f}\text{(x)}\text{dx}=\int\limits_{0}^{\text{a}}\text{f}\text{(a}-\text{x})\text{dx}=\bigg]$
$\Rightarrow\ \text{I}=\int\limits_{0}^{1}(1-\text{x})(1-1+\text{x})^{\text{n}}\ \text{dx}=\int\limits_{0}^{1}(1-\text{x})\text{x}^{\text{n}}\ \text{dx}=\int^{1}_{0}(\text{x}^{\text{n}}-\text{x}^{\text{n}+1})\text{dx}$
$\Rightarrow\ \ \text{I}=\bigg(\frac{\text{x}^{\text{n}+1}}{\text{n}+1}-\frac{\text{x}^{\text{n}+2}}{\text{n}+2}\bigg)^{1}_{0}=\frac{1}{\text{n}+1}-\frac{1}{\text{n}+2}-(0-0)=\frac{\text{n}+2-\text{n}-1}{(\text{n}+1)(\text{n}+2)}=\frac{1}{\text{(n}+1)(\text{n}+2)}$

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 "2 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Write a vector in the direction of vector $5\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}$ which has magnitude of 8 unit.

Check the injectivity and surjectivity of the below function:
$f : Z \rightarrow Z$ given by $f(x) = x^2$

A random variable has the following probability distribution:

$X = X_i$	1	2	3	4
$P(X = X_i)$	k	2k	3k	4k

Write the value of $\text{P}(\text{X}\geq3).$

Matrix $\text{A}=\begin{bmatrix}0&2\text{b}&-2\\3&1&3\\3\text{a}&3&-1 \end{bmatrix}$ is given to be symmetric, find values of a and b.

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector $3\hat i + 2\hat j - 2\hat k$.

Find the angle between the lines $\vec{\text{r}}=\big(2\hat{\text{i}}-5\hat{\text{j}}+\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\big)$ and $\vec{\text{r}}=7\hat{\text{i}}-6\hat{\text{k}}+\mu\big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\big).$

If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the given vectors are non-coplanar:
$2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}},\ \vec{\text{a}}+\vec{\text{b}}-2\vec{\text{c}}$ and $\vec{\text{a}}+\vec{\text{b}}-3\vec{\text{c}}$

Find the equation of the curve which passes through the point (3, -4) and has slope $\frac{2\text{y}}{\text{x}}$ at any point (x, y) on it.

Verify that the function $y = x^2 + 2x + C$ (explicit or implicit) is a solution of differential equation $y' - 2x - 2 = 0$

Write the function in the simplest form: ${\tan ^{ - 1}}\frac{{\sqrt {1 + {x^2}} - 1}}{x},x \ne 0$.