Download App

Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]3 Marks
Question
Evaluate: $\int_0^3 x^2(3-x)^{\frac{5}{2}} d x$
✓

Answer

$\text { Let } I =\int_0^3 x^2(3-x)^{\frac{5}{2}} d x$
$=\int_0^3(3-x)^2[3-(3-x)]^{\frac{5}{2}} d x \quad \ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
$=\int_0^3\left(9-6 x+x^2\right) x^{\frac{5}{2}} d x$
$=\int_0^3\left(9 x^{\frac{5}{2}}-6 x^{\frac{7}{2}}+x^{\frac{9}{2}}\right) d x$
$=9 \int_0^2 x^{\frac{5}{2}} d x-6 \int_0^3 x^{\frac{7}{2}} d x+\int_0^3 x^{\frac{9}{2}} d x$
$=9\left[\frac{x^{\frac{7}{2}}}{\frac{7}{2}}\right]_0^3-6\left[\frac{x^{\frac{9}{2}}}{\frac{9}{2}}\right]_0^3+\left[\frac{x^{\frac{11}{2}}}{\frac{11}{2}}\right]_0^3$
$=\frac{18}{7}\left[(3)^{\frac{7}{2}}-0\right]-\frac{12}{9}\left[(3)^{\frac{9}{2}}-0\right]+\frac{2}{1}\left[(3)^{\frac{11}{2}}-0\right]$
$=\left[\frac{18}{7}-\left(\frac{12}{9} \times 3\right)+\left(\frac{2}{11} \times 9\right)\right](3)^{\frac{7}{2}}$
$=\left(\frac{198-308+126}{77}\right)(3)^{\frac{7}{2}}$
$\therefore I =\frac{16}{77}(3)^{\frac{7}{2}}$

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 "Solve the Following Question.(3 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Solve the following differential equations:
$\text{xy}\frac{\text{dy}}{\text{dx}}=1+\text{x + y + xy}$
→
Integrate the following functions w.r.t x:

$3^{\cos ^2 x} \sin 2 x$

→
Find $\frac{d y}{d x}$ if

$x=\left(t+\frac{1}{t}\right)^a, y=a^{t+\frac{1}{t}}$, where $a>0, a \neq 1$ and $t \neq 0$

→
If $\log _5\left(\frac{x^4+y^4}{x^4-y^4}\right)=2$, show that $\frac{d y}{d x}=-\frac{12 x^3}{13 y^3}$
→
If in parallelogram $A B C D$, diagonal vectors are $\overline{A C}=2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $\overline{B D}=$

$-6 \hat{i}+7 \hat{j}-2 \hat{k}$, then find the adjacent side vectors $\overline{A B}$ and $\overline{A D}$

→
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than $4$ appears on at least one die.
→
Solve the following equation for x:
$3\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}-4\cos^{-1}\frac{1-\text{x}^2}{1+\text{x}^2}+2\tan^{-1}\frac{2\text{x}}{1-\text{x}^2}=\frac{\pi}{3}$
→
Evalute the following integrals:
$\int\frac{\cos\text{x}}{\cos(\text{x}-\text{a})}\text{dx}$
→
Evaluate the following :

$\int \frac{1}{\cos 2 x+3 \sin ^2 x} \cdot d x$

→
Show that the line joining the origin to the point $(2, 1, 1)$ is perpendicular to the line determined by the points $(3, 5, -1)$ and $(4, 3, -1).$
→