Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Integrate the function in Exercise:
$\text{e}^\text{x}\Bigg(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Bigg)$
✓

Answer

$\text{e}^\text{x}\Bigg(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Bigg)$
$=\text{e}^\text{x}\Bigg(\frac{\sin^2\frac{\text{x}}{2}+\cos^2\frac{\text{x}}{2}+2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}\Bigg)$
$=\frac{\text{e}^\text{x}\Big(\sin\frac{\text{x}}{2}+\cos\frac{\text{x}}{2}\Big)^2}{2\cos^2\frac{\text{x}}{2}}$
$=\frac{1}{2}\text{e}^\text{x}.\Bigg(\frac{\sin\frac{\text{x}}{2}+\cos\frac{\text{x}}{2}}{\cos\frac{\text{x}}{2}}\Bigg)^2$
$=\frac{1}{2}\text{e}^\text{x}\Bigg[\tan\frac{\text{x}}{2}+1\Bigg]^2$
$=\frac{1}{2}\text{e}^\text{x}\Bigg[1+\tan\frac{\text{x}}{2}\Bigg]^2$
$=\frac{1}{2}\text{e}^\text{x}\Bigg[1+\tan^2\frac{\text{x}}{2}+2\tan\frac{\text{x}}{2}\Bigg]$
$=\frac{1}{2}\text{e}^\text{x}\Bigg[\sec^2\frac{\text{x}}{2}+2\tan\frac{\text{x}}{2}\Bigg]$
$\frac{\text{e}^\text{x}(1+\sin\text{x})\text{dx}}{(1+\cos\text{x})}=\text{e}^\text{x}\Bigg[\frac{1}{2}\sec^2\frac{\text{x}}{2}+\tan\frac{\text{x}}{2}\Bigg]$
Let $\tan\frac{\text{x}}{2}=\text{f(x)}\Rightarrow\ \text{f}'(\text{x})=\frac{1}{2}\sec^2\frac{\text{x}}{2}$
It is known that, $\int\text{e}^\text{x}\{\text{f}(\text{x})+\text{f}'(\text{x})\}\text{dx}=\text{e}^\text{x}\ \ \text{f}(\text{x})+\text{C}$
From equation (1), we obtain
$​​​​​​\int\frac{\text{e}^\text{x}(1+\sin\text{x})}{(1+\cos\text{x})}\text{dx}=\text{e}^\text{x}\tan\frac{\text{x}}{2}+\text{C}$

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 "5 Marks Questions" 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

The length x of a rectangle is decreasing at the rate of 5cm/ minute and the width y is increasing at the rate of 4cm/ minute. When x = 8cm and y = 6cm, find the rates of change of:
  1. The perimeter.
  2. The area of the rectangle.
→
Show that the following systems of linear equations has infinite number of solutions and solve:
$2x + y - 2z = 0,$
$x - 2y + z = -2,$
$5x - 5y + z = -2$
→
An urn contains 4 red and 3 blue balls. Find the probability distribution of the number of blue balls in a random draw of 3 balls with replacement.
→
A and B take turns in throwing two dice, the first to throw 10 being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio 12 : 11.
→
Solve the following differential equations:
$(\text{x}-1)\frac{\text{dy}}{\text{dx}}=2\text{x}^3\text{y}$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^3- 12x^2 + 36x + 17$
→
Find the area bounded by the cutve $y = 4 - x^2$​​​​​​​ and the line $y = 0, y = 3.$
→
Show that the relation $R,$ defined on the set $A$ of all polygons as $R = \{(P_1, P_2): P_1$ and $P_2$ have same number of sides$\},$ is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3, 4$ and $5?$
→
Find the equation of a curve passing through the point (0, 0) and whose differential equation is $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}\sin\text{x.}$
→
Evaluate the following integrals as limit of sum:
$\int\limits^{2}_{0}\big(\text{x}^2+2\big)\text{dx}$
→