Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat 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" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following functions with respect to x:
$\sin(3\text{x}+5)$
→
Find a vector of magnitude 49, which is perpendicular to both the vectors $2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}$ and $3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}.$
→
If $\vec{a}+\vec{b}, \vec{b}+\vec{c}$ and $\vec{c}+\vec{a}$ are coplanar, then show that $\vec{a}, \vec{b}$ and $\vec{c}$ will be coplanar.
→
Find the angle between the plane:
x - y + z = 5 and x + 2y + z = 9
→
Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=3$

Function $\text{y}=\text{e}^\text{-x}+2$

→
If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (-4, 3, -6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
→
Differentiate w.r.t. $x$ of $y$$
=x+\frac{1}{x+\frac{1}{x+\frac{1}{x+\ldots \ldots \infty}}}
$
→
Prove the following results:
$4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239}=\frac{\pi}{4}$
→
Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows:
(a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R.
Find the identity element in A.
→
From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that both of them are kings.
→