Evaluate the following integrals: ^2_0x 2-x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^2_0\text{x}\sqrt{2-\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\limits^2_0\text{x}\sqrt{2-\text{x}}\text{ dx}$
$=\int\limits^2_0(2-\text{x})\sqrt{2-2+\text{x}}\text{ dx}$
$=\int\limits^2_0(2-\text{x})\sqrt{\text{x}}\text{ dx}$
$=\int\limits^2_0\big(2\sqrt{\text{x}}-\text{x}\sqrt{\text{x}}\big)\text{dx}$
$=\int\limits^2_0\Big(2\text{x}^{\frac{1}{2}}-\text{x}^{\frac{3}{2}}\Big)\text{dx}$
$=\Bigg[2\frac{\text{x}^{\frac{3}{2}}}{\frac{3}{2}}-\frac{\text{x}^{\frac{5}{2}}}{\frac{5}{2}}\Bigg]$
$=\bigg[\frac{4}{3}\text{x}^{\frac{3}{2}}-\frac{2}{5}\text{x}^{\frac{5}{2}}\bigg]^2_0$
$=\frac{8\sqrt{2}}{3}-\frac{8\sqrt{2}}{5}$
$=\frac{16\sqrt{2}}{15}$

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 Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Let $A$ and $B$ be two independent events such that $P(A) = p_1$ and $P(B) = p_2.$ Describe in words the events whose probabilities are: $1 - (1 - p_1)(1 - p_2).$

Find the value of k in this question, so that the function f is continuous at the indicated point:
$\text{f(x)}=\begin{cases}3\text{x}-8,&\text{if x}\leq5\\2\text{k},&\text{if x}>5\end{cases}$ at x = 5.

On Q, the set of all rational numbers, * is defined by $\text{a}\ ^*\ \text{b}=\frac{\text{a}-\text{b}}{2},$ shown that * is no associative.

Differentiate the following w.r.t.x: $\sin(\tan^{-1}\text{e}^{-\text{x}})$

Three machines $E_1, E_2, E_3$ in a certain factory produce $50\%, 25\%$ and $25\%,$ respectively, of the total daily output of electric bulbs. It is known that $4\%$ of the tubes produced one each of the machines $E_{1 }$ and $E_2$ are defective, and that $5\%$ of those produced on $E_3$ are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.

Evaluate the following integrals:$\int\sec^{-1}\sqrt{\text{x}}\text{dx}$

Show that $f: [–1, 1] \rightarrow R$, given by $f(\text{x})=\frac{\text{x}}{(\text{x}+2)}$ is one-one. Find the inverse of the function $f:[-1, 1] \rightarrow $ Range $f.$
(Hint: For $\text{y}\in\text{Range f},\text{y}=f(\text{x})=\frac{\text{x}}{\text{x}+2},$ for some $x$ in $[-1, 1],$ i.e., $\text{x}=\frac{2\text{y}}{(1-\text{y})}$)

Evaluate $\int e^{2 x}\left(\frac{1-\sin 2 x}{1-\cos 2 x}\right) d x$

Solve the following differential equations:$\frac{\text{dr}}{\text{dt}}=-\text{rt, r}(0)=\text{r}_{0}$

There are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.