Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{1}{5+7\cos\text{x}+\sin\text{x}}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{5+7\cos\text{x}+\sin\text{x}}\ \text{dx}$
Putting $\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\text{ and }\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
Now,
$\text{I}=\int\frac{1}{5+\frac{7\Big(1-\tan\frac{\text{x}}{2}\Big)}{\Big(1+\tan^2\frac{\text{x}}{2}\Big)}+\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)}\text{ dx}$
$=\int\frac{\Big(1+\tan^2\frac{\text{x}}{2}\Big)}{5\Big(1+\tan^2\frac{\text{x}}{2}\Big)+7-7\tan^2\frac{\text{x}}{2}+2\tan\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{-2\tan^2\frac{\text{x}}{2}+12+2\tan\frac{\text{x}}{2}}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$\text{I}=\int\frac{2\text{dt}}{-2\text{t}^2+12+2\text{t}}$
$=-\int\frac{\text{dt}}{\text{t}^2-\text{t}-6}$
$=-\int\frac{\text{dt}}{\text{t}^2-2\text{t}\Big(\frac{1}{2}\Big)+\Big(\frac{1}{2}\Big)^2-\Big(\frac{1}{2}\Big)^2-6}$
$=-\int\frac{\text{dt}}{\Big(\text{t}-\frac{1}{2}\Big)^2-\Big(\frac{5}{2}\Big)^2}$
$=-\frac{1}{2\Big(\frac{5}{2}\Big)}\log\Bigg|\frac{\text{t}-\frac{1}{2}-\frac{5}{2}}{\text{t}-\frac{1}{2}+\frac{5}{2}}\Bigg|+\text{C}$
$=-\frac{1}{5}\log\Big|\frac{\text{t}-3}{\text{t}+2}\Big|+\text{C}$
$\text{I}=\frac{1}{5}\log\Bigg|\frac{\tan\frac{\text{x}}{2}+2}{\tan\frac{\text{x}}{2}-3}\Bigg|+\text{C}$

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 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

Differentiate $\tan^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$ with respect to $\sqrt{1-\text{x}^2},$ if -1 < x < 1.
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\text{ dx}\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\ \text{dx}$
→
If f'(x) = x + b, f'(1) = 5, f'(2) = 13, find f'(x).
→
By examining the chest X-ray, probability that T.B. is detected when a person is actually suffering is 0.99. The probability that the doctor diagnoses incorrectly that a person has T.B. on the basic of X-ray is 0.001. In a certain city 1 in 1000 persons suffers from T.B. A person is selected at random is diagnosed to have T.B. what is the chance that he actually has T.B.?
→
If $\text{y}=(\text{x}-1)\log(\text{x}-1)-(\text{x}+1)\log(\text{x}+1)$ prove that $\frac{\text{dy}}{\text{dx}}=\log\Big(\frac{\text{x}-1}{1+\text{x}}\Big)$
→
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is $\frac{8}{{27}}$ of the volume of the sphere.
→
Evaluate:
$\int\frac{1}{\sin^{4}\text{x} +\sin^{2}\text{x}\cos^{2}\text{x}+\cos^{4}\text{x}}\text{dx}$
→
If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix},$ show that $A^2 - 5A + 7I_2 = 0$.
→
If R is the largest equivalence relation on a set A and S is any relation on A, then:
  1. $\text{R}\subset\text{S}$
  2. $\text{S}\subset\text{R}$
  3. $\text{R = S}$
  4. None of these.
→
Solve the following differential equation:
$\text{y}^2\frac{\text{dx}}{\text{dy}}+\text{x}-\frac{1}{\text{y}}=0$
→