Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS3 Marks
Question
Evaluate the definite integral in Exercise:
$\int\limits^{\frac{\pi}{4}}_0\frac{\sin\text{x}\cos\text{x}}{\cos^{4}\text{x}+\sin^{4}\text{x}}\text{dx}$
✓

Answer

$\text{Let I}=\int^{\frac{\pi}{4}}\limits_{0}\frac{\sin\text{x}\cos\text{x}}{\cos^{4}\text{x}+\sin^{4}\text{x}}\text{dx}$
$\Rightarrow\text{ I}=\int^{\frac{\pi}{4}}\limits_{0}\frac{\frac{(\sin\text{x}\cos\text{x)}}{\cos^{4}\text{x}}}{\frac{\cos^{4}\text{x}+\sin^{4}\text{x}}{\cos^{4}\text{x}}}\text{dx}$
$\Rightarrow\text{ I}=\int^{\frac{\pi}{4}}\limits_{0}\frac{\tan\text{x}\sec^{2}\text{x}}{1+\tan^{4}\text{x}}\text{dx}$
$\text{Let}\ \tan^{2}\text{x}=\text{t}\ \Rightarrow2\tan\text{x}\text{dx}=\text{dt}$
when $\text{x}=0,\text{t}=0$ and when $\text{x}=\frac{\pi}{4},\text{t}=1$
$\therefore\ \text{I}=\frac{1}{2}\int^{1}\limits_{0}\frac{\text{dt}}{1+\text{t}^{2}}$
$=\frac{1}{2}\Big[\tan^{-1}\text{t}\Big]^{1}_{0}$
$=\frac{1}{2}\Big[\tan^{-1}-\tan^{-1}0\Big]$
$=\frac{1}{2}\Big[\frac{\pi}{4}\Big]$
$=\frac{\pi}{8}$

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

Give an example of a relation which is,Symmetric but neither reflexive nor transitive.
→
Let $R_0$ denote the set of all non $-$ zero real numbers and let $A = R_0 \times R_0$. If $'*'$ is a binary operation on adefined by,
$(a, b) * (c, d) = (ac, bd)$ for all $(a, b), (c, d) \in A$
Show that '*' is both commutative and associative on $A$.
→
Find the area of the triangle formed by O, A, B when $\overrightarrow{\text{OA}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\overrightarrow{\text{OB}}=-3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}.$
→
Evaluate the following:
$\int\frac{(1+\cos\text{x})}{\text{x}+\sin\text{x}}\text{dx}$
→
Find the coordinates of the point where the line$\frac{\text{x + 1}}{2}=\frac{\text{y + 2}}{3}=\frac{\text{z + 3}}{4}$
meets the plane x + y + 4z = 6.
→
Prove the following:
$2\tan^{-1}\frac{1}{2}+\tan^{-1}\frac{1}{7}=\tan^{-1}\frac{31}{17}$
→
Discuss the continuity of the function $\text{f(x)}=\begin{cases}\frac{\text{x}}{|\text{x}|},&\text{x}\neq0\\0,&\text{x}=0\end{cases}$
→
Solve the following equation for x:
$\tan^{-1}(2+\text{x})+\tan^{-1}(2-\text{x})=\tan^{-1}\frac{2}{3},$ where $\text{x}<-\sqrt3$ or, $\text{x}>\sqrt3$
→
For each of the differential equations given in find the general solution:
$(\text{x}+3\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}(\text{y}>0)$
→
If the function $f: R\rightarrow R$ be given by $f(x) = x^2 + 2$ and $g: R \rightarrow R$ be given by $g(x) = \frac{\text{x}}{\text{x} - 1 },\text{x}\neq1 ,$ find fog and gof and hence find fog $(2)$ and gof $(–3).$
→