Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following intregals:
$\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$
$\text{I}=\int\frac{2}{2+2\sin\text{x}\cos\text{x}}\ \text{dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\text{I}=\int\frac{\frac{1}{\cos^2\text{x}}}{\frac{1}{\cos^2\text{x}}+\frac{\sin\text{x}\cos\text{x}}{\cos^2\text{x}}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{\sec^2\text{x}+\tan\text{x}}\ \text{dx}$
$\text{I}=\int\frac{\sec^2\text{x}}{1+\tan^2\text{x}+\tan\text{x}}\ \text{dx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{x dx}=\text{dt}$
$\text{I}=\int\frac{\text{dt}}{\text{t}^2+\text{t}+1}$
$=\int\frac{\text{dt}}{\text{t}^2+2\text{t}\Big(\frac{1}{2}\Big)+\Big(\frac{1}{2}\Big)^2-\Big(\frac{1}{2}\Big)^2+1}$
$\text{I}=\int\frac{\text{dt}}{\Big(\text{t}+\frac{1}{2}\Big)^2+\Big(\frac{\sqrt{3}}{2}\Big)^2}$
$=\frac{1}{\frac{\sqrt{3}}{2}}\tan^{-1}\Bigg(\frac{\text{t}+\frac{1}{2}}{\frac{\sqrt{3}}{2}}\Bigg)+\text{C}$
$=\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{2\text{t}+1}{\sqrt{3}}\Big)+\text{C}$
$\text{I}=\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{2\tan\text{x}+1}{\sqrt{3}}\Big)+\text{C}$

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 "4 Marks" from Indefinite Integrals→Indefinite 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 factory owner purchases two types of machines, A and B, for his factory. The requirements and limitations for the machines are as follows:
 
Area occupied by the
machine
Labour force for each
machine
Daliy outputin
units
Machines
1000 sp.m
12 mem
60
Machines
1200 sp.m
8 mem
40
He has an area of 7600 sq. m available and 72 skilled men who can operate the machines.
How many machines of each type should he buy to maximize the daily output?
→
For each of the differential equations given in find a particular solution satisfying the given condition:
$\frac{\text{dy}}{\text{dx}}+2\text{y}\tan\text{x}=\sin\text{x};\text{y}=0\ \text{when x}=\frac{\pi}{3}$
→
Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$
→
Evaluate the following integrals as limit of sum:
$\int\limits^3_{2}\big(2\text{x}^2+1\big)\text{ dx}$
→
If ey= yx, prove that $\frac{\text{dy}}{\text{dx}}=\frac{(\log\text{y})^2}{\log\text{y}-1}$
→
$\text{If x = a}\sin 2\text{t} (1 + \cos\text{2t) and y = b}\cos\text{2t (1} - \cos \text{2t)}, $ find the values of $\frac{\text{dy}}{\text{dx}} \text{at t} = \frac{\pi}{4} \text{and t} \frac{\pi}{3}.$
→
Integrate the rational function in exercise:
$\frac{1}{(\text{e}^\text{x}-1)}$
[Hint: Put ex = t]
→
Using the method of interation, find the area of the region bounded by the following lines: 
3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0.
→
Determine the values of a, b, c for which the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin\text{(a}+1)\text{x}+\sin\text{x}}{\text{x}}, &\text{for}\text{ x}<0,&\\\text{ c},&\text{for x}=0\\\frac{\sqrt{\text{x}+\text{bx}^2}-\sqrt{\text{x}}}{\text{bx}^\frac{3}{2}},&\text{for x}>0\end{cases}$ is continuous at x = 0.
→
Prove that:
$\begin{vmatrix} 1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}=\begin{vmatrix} 1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\\1&\text{c}&\text{c}^2\end{vmatrix}$
→