Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS3 Marks
Question
By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\frac{\cos^{5}\text{x}\ \text{dx}}{\sin^{5}\text{x}+\cos^{5}\text{x}}$
✓

Answer

$\text{Let}\text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\cos^{5}\text{x}\ }{\sin^{5}\text{x}+\cos^{5}\text{x}}\text{dx}$ $\Rightarrow\ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\cos^{5}\bigg(\frac{\pi}{2}-\text{x}\bigg)}{\sin^{5}\bigg(\frac{\pi}{2}-\text{x}\bigg)+\cos^{5}\bigg(\frac{\pi}{2}-\text{x}\bigg)}\text{dx}\ \ \bigg[\because\int\limits_{0}^{\text{a}}\text{f}\text{(x)}\ \text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{(a}-\text{x)}\text{dx}=\bigg]$ $\Rightarrow\ \ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{5}\text{x}}{\cos^{5}\text{x}+\sin^{5}\text{x}}\text{dx}$Adding eq. (i) and (ii),
$21=\int\limits_{0}^{\frac{\pi}{2}}\bigg(\frac{\cos^{5}\text{x}}{\sin^{5}\text{x}+\cos^{5}\text{x}}+\frac{\sin^{5}\text{x}}{\cos^{5}\text{x}+\sin^{5}\text{x}}\bigg)\text{dx}=\int\limits_{0}^{\frac{\pi}{2}}\bigg(\frac{\cos^{5}\text{x}+\sin^{5}\text{x}}{\sin^{5}\text{x}+\cos{5}\text{x}}\bigg)\text{dx}$ $\Rightarrow\ \ 21=\int\limits_{0}^{\frac{\pi}{2}}1\ \text{dx}=\bigg(\text{x}^{\frac{\pi}{2}}_{0}\bigg)\ \Rightarrow21=\frac{\pi}{2}\ \Rightarrow\text{I}=\frac{\pi}{4}$

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

Let $A = \{1, 2, 3\},$ and let $R_1 = \{(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)\}.$ Find whether or not the relations $R_{1 }$ on $A$ is:
  1. Reflexive.
  2. Symmetric.
  3. Transitive.
→
Find the distance of the point P(-1, -5, -10) from the point of intersection of the line joining the points A(2, -1, 2) and B(5, 3, 4) with the plane x - y + z = 5.
→
Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.
→
If $\text{f}\text{(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},& \text{when}\text{ x}\neq0 \\1,&\text{when} \text{ x}=0\end{cases}$ Find whether f(x) is continuous at x = 0.
→
Solve the following differential equations:$2\text{x}\frac{\text{dy}}{\text{dx}}=3\text{y},\text{y}(1)=2$
→
Find the real solutions of the equation $\tan^{-1}\sqrt{\text{x}(\text{x}+1)}+\sin^{-1}\sqrt{\text{x}^2+\text{x}+1}=\frac{\pi}{2}.$
→
$\int\frac{1}{\cos^2\text{x}(1-\tan\text{x})^2}\text{dx}$
→
If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution.
→
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 the following integrals:
$\int\cot \text{x}. \text{log}\ \sin\text{x dx}$
→