Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS3 Marks
Question
Evaluate the integral in Exercise:
$\int\limits^{\frac{\pi}{2}}_{0}\sqrt{\sin\phi}\cos^{5}\phi\text{ d }\phi$
✓

Answer

$\text{Let}\ \text{I}=\int\limits^\frac{\pi}{2}_{0}\sqrt{\sin\phi}\cos^{5}\phi\ \text{d}\ \phi$
$\text{putting}\ \sin\phi=\text{t}\ \Rightarrow\ \cos\phi=\frac{\text{dt}}{\text{d}\phi}\ \Rightarrow\ \cos\phi\ \text{d}\ \phi=\text{dt}$
To change the limits of integration from   $\phi \text{to}\ \text{t}$ 
$\text{when}\phi=0,\text{t}=\sin\phi=\sin0^{\text{o}}=0$
$\text{when}\phi=\frac{\pi}{2},\text{t}=\sin\phi=\sin\frac{\pi}{2}=1$
$\therefore$ From eq.(i),$\ \text{I}=\int\limits_{0}^\frac{\pi}{2}\sqrt{\sin}\phi\cos^{5}\phi\cos\phi\ \text{d}\ \phi=\int^\frac{\pi}{2}_{0}\sqrt{\sin\phi}(\cos^{2}\phi)^{2}\cos\phi\ \text{d}\ \phi$
$=\int\limits_{0}^\frac{\pi}{2}\sqrt{\sin}\phi(1-\sin^{2}\phi)^{2}\cos\phi\ \text{d}\ \phi=\int\limits_{0}^{1}\sqrt{\text{t}}(1-\text{t}^{2})^{2}\text{dt}=\int\limits_{0}^{1}\text{t}^\frac{1}{2}(1+\text{t}^{4}-2\text{t}^{2})\text{dt}$
$=\int\limits_{0}^{1}\bigg(\text{t}^\frac{1}{2}+\text{t}^{\frac{1}{2}+4}-2\text{t}^{\frac{1}{2}+2}\bigg)\text{dt}=\int\limits_{0}^{1}\bigg(\text{t}^\frac{1}{2}+\text{t}^\frac{9}{2}-2\text{t}^\frac{5}{2}\bigg)\text{dt}=\int\limits_{0}^{1}\text{t}^\frac{1}{2}\text{dt}+\int\limits_{0}^{1}\text{t}^\frac{9}{2}\text{dt}-2\int\limits_{0}^{1}\text{t}^\frac{5}{2}\text{dt}$
$=\frac{\bigg(\text{t}^\frac{3}{2}\bigg)^{1}_{0}}{\frac{3}{2}}+\frac{\bigg(\text{t}^\frac{11}{2}\bigg)^{1}_{0}}{\frac{11}{2}}+\frac{\bigg(\text{t}\frac{7}{2}\bigg)^{1}_{0}}{\frac{7}{2}}=\frac{2}{3}(1-0)+\frac{2}{11}(1-0)-\frac{4}{7}(1-0)=\frac{2}{3}+\frac{2}{11}-\frac{4}{7}$
$=\frac{154+42-132}{231}=\frac{64}{231}$

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

Similar questions

Bag $I$ contains $3$ white and $4$ black balls, while Bag $II$ contains $5$ white and $3$ black balls. One ball is transferred at random from Bag $I$ to Bag $II$ and then a ball is drawn at random from Bag $II$. The ball so drawn is found to be white. Find the probability that the transferred ball is also white.
→
A bill for Rs. 7650 was drawn on 8th March 2005 at 7 months. It was discounted on 18 May 2005 and the holder of the bill received Rs. 7497. What rate of interest did the banker charge?
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}1&-3&2\\4&-1&3\\3&5&2\end{vmatrix}$
→
If $\text{f(x)}=\sqrt{\text{x}+3}$ and $g(x) = x^2 + 1$ be two real functions, then find fog and gof.
→
Evaluate the following definite integrals:
$\int_{1}^\limits{2}\log\text{x}\text{ dx}$
→
Show that the solution set of the linear constraints is empty $x-2 y \geq 0,2 x-y \leq-2, x \geq 0$ and $y \geq 0$
→
Find the second $-$ order derivatives of the function $\tan^{-1}x$
→
Solve the following equation for x:
$\cos^{-1}\Big(\frac{\text{x}^2-1}{\text{x}^2+1}\Big)+\frac{1}{2}\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)=\frac{2\pi}{3}$
→
Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$
→
Write the projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}}$ when $\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$
→