Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate:
$\int\limits^{\pi/4}_{0}\bigg(\frac{\sin \text{x} +\cos \text{x}}{3 + \sin 2\text{x}}\bigg)\text{dx}$
✓

Answer

$\text{I} = -\int\limits^\frac{\pi}{4}_{0} \frac{\sin \text{x} + \cos \text{x}}{(\sin \text{x} - \cos \text{x})^{2} - 2^{2}}\text{dx}$
$\text{Put} \sin \text{x} - \cos \text{x} = \text{t} \Rightarrow \text{t} = \text{-1 to } 0$
$(\cos \text{x} + \sin {\text{x)}} \text{dx = dt} $
$\text{I} = - \int\limits^{0}_{-1}\frac{\text{dt}}{\text{t}^{2} - 2^{2}}$
$= -\frac{1}{4}\log\bigg|\frac{\text{t} - 2}{\text{t} + 2}\bigg|\bigg]^{0}_{-1}$
$= -\frac{1}{4}\big\{0 - \log 3\big\}$
$= \frac{1}{4}\log3$

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 "4 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

A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.
→
Examine the differentiability of f, where f is defined by:
$\text{f(x)}=\begin{cases}1+\text{x},&\text{if x}\leq2\\5-\text{x},&\text{if x}>2\end{cases}$
at x = 2.
→
An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.
→
If $\text{A}=\begin{bmatrix}0&\text{c}&-\text{b}\\-\text{c}&0&\text{a}\\\text{b}&-\text{a}&0\end{bmatrix}$ and $\text{B}=\begin{bmatrix}\text{a}^2&\text{ab}&\text{ac}\\\text{ab}&\text{b}^2&\text{bc}\\\text{ac}&\text{bc}&\text{c}^2\end{bmatrix},$ show that AB = BA = O3 × 3
→
Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\text{y}=\cos^{-1}\Bigg(\frac{2\text{x}}{1+\text{x}^{2}}\Bigg), -1<\text{x}<1$
→
Form the differential equation of the family of curve represented by y2 = (x - c)3
→
Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$
→
Show that $\tan\Big(\frac{1}{2}\sin^{-1}\frac{3}{4}\Big)=\frac{4-\sqrt{7}}{3}$ and justify why the other value $\frac{4+\sqrt{7}}{3}$ is ignored?
→
Using matrices, solve the following system of equations:

4x + 3y + 2z = 60.

x + 2y + 3z = 45.

6x + 2y + 3z = 70.

→
A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman's time.
  1. What number of rackets and bats must be made if the factory is to work at full capacity?
  2. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.
→