Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{1}{\sin\text{x}-\sqrt{3}\cos\text{x}}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{\sin\text{x}-\sqrt{3}\cos\text{x}}\ \text{dx}$
Putting $\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\text{ and }\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\Rightarrow\text{I}=\int\frac{1}{\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}-\sqrt{3}\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}}\ \text{dx}$
$=\int\frac{1+\tan^2\frac{\text{x}}{2}}{2\tan\frac{\text{x}}{2}-\sqrt{3}+\sqrt{3}\tan^2\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{\sqrt{3}\tan^2\frac{\text{x}}{2}+2\tan\frac{\text{x}}{2}-\sqrt{3}}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\Rightarrow\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$\Rightarrow\sec^2\frac{\text{x}}{2}\text{dx}=2\text{dt}$
$\therefore\text{I}=2\int\frac{\text{dt}}{\sqrt{3}\text{t}^2+2\text{t}-\sqrt{3}}$
$=\frac{2}{\sqrt{3}}\int\frac{\text{dt}}{\text{t}^2+\frac{2}{\sqrt{3}}\text{t}-1}$
$=2\int\frac{\text{dt}}{\text{t}^2+\frac{2}{\sqrt{3}}\text{t}+\Big(\frac{1}{\sqrt{3}}\Big)^2-1-\Big(\frac{1}{\sqrt{3}}\Big)^2}$
$=2\int\frac{\text{dt}}{\Big(\text{t}+\frac{1}{\sqrt{3}}\Big)^2-\Big(\frac{2}{\sqrt{3}}\Big)^2}$
$=\frac{2}{2\times\frac{2}{\sqrt{3}}}\log\Bigg|\frac{\text{t}+\frac{1}{\sqrt{3}}-\frac{2}{\sqrt{3}}}{\text{t}+\frac{1}{\sqrt{3}}+\frac{2}{\sqrt{3}}}\Bigg|+\text{C}$
$=\frac{\sqrt{3}}{2}\log\Bigg|\frac{\tan\frac{\text{x}}{2}-\frac{1}{\sqrt{3}}}{\tan\frac{\text{x}}{2}+\frac{3}{\sqrt{3}}}\Bigg|+\text{C}$
$=\frac{\sqrt{3}}{2}\log\Bigg|\frac{\sqrt{3}\tan\frac{\text{x}}{2}-1}{\sqrt{3}\tan\frac{\text{x}}{2}+3}\Bigg|+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The radius of a cylinder is increasing at the rate 2cm/ sec. and its altitude is decreasing at the rate of 3cm/ sec. Find the rate of change of volume when radius is 3cm and altitude 5cm.
→
Find the equation of the plane through the points $(3, 4, 1)$ and $(0, 1, 0)$ and parallel to the line $\frac{\text{x}+3}{2}=\frac{\text{y}-3}{7}=\frac{\text{z}-2}{5}.$
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}\text{ on }\Big[0,\frac{\pi}{6}\Big]$
→
If $l_i, m_i, n_i; i = 1, 2, 3$ denotes the direction cosines of three mutually perpendicular vectors in space, prove that $AA^T = I$, where
$\text{A}=\begin{bmatrix}\text{l}_1&\text{m}_1&\text{n}_1\\\text{l}_2&\text{m}_2&\text{n}_2\\\text{l}_3&\text{m}_3&\text{n}_3 \end{bmatrix}.$
→
If $\text{A}=\begin{bmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{bmatrix},$ find $(A^T)^{-1}.$
→
Differentiate $\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big),$ if 0 < x < 1.
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x e }^\text{x}\log\text{x}+\text{e}^\text{x}}{\text{x}\cos\text{y}}$
→
Find the intervals in which $\text{f}(\text{x})=\log(1+\text{x})-\frac{\text{x}}{1+\text{x}}$ is increasing or decreasing.
→
Solve the following differential equation
$\sqrt{\text{a}+\text{x}}\text{dy}+\text{x dx}=0$
→
Solve the following systems of linear equations by cramer's rule:
2x - y = 17,
3x + 5y = 6
→