Evaluate: 1 x+3 x d x - Vidyadip

Question

Evaluate: $\int \frac{1}{\sin x+\sqrt{3} \cos x} d x$

✓

Answer

$(a) :$ Let $I=\int \frac{1}{\sin x+\sqrt{3} \cos x} d x$
$=\frac{1}{2} \int \frac{d x}{\frac{1}{2} \sin x+\frac{\sqrt{3}}{2} \cos x}$
$\Rightarrow I=\frac{1}{2} \int \frac{1}{\sin \left(x+\frac{\pi}{3}\right)} d x=\frac{1}{2} \int \operatorname{cosec}\left(x+\frac{\pi}{3}\right) d x$
$\Rightarrow I=\frac{1}{2} \log \left|\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right|+C$
${\left[\because \int \operatorname{cosec} x d x=\log \left|\tan \frac{x}{2}\right|+C\right]}$

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 "M.C.Q (1 Marks)" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6 then $\text{P}(\text{A}\cup\text{B})=$

0.24
0.3
0.48
0.96

The objective function Z = 4x + 3y can be maximised subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6, x, y ≥ 0

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is:

Identify relation.
Reflexive.
Symmetric.
Antisymmetric.

The equation of the normal to the curve $\text{y}=\text{x}+\sin\text{x}\cos\text{x}\text{ at }\text{x}=\frac{\pi}{2}$ is

$\text{x}= 2$
$\text{x}=\pi$
$\text{x}+\pi=0$
$2\text{x}=\pi$

If a line makes angles $Q_1, Q_{21}$ and $Q_3$ respectively with the coordinate axis then the value of $\cos^2 \text{Q}_{1} + \cos^2 \text{Q}_{2} + \cos^2 \text{Q}_{3}:$

For real numbers x and y, define xRy if $\text{x}-\text{y}+\sqrt{2}$ is an irrational number. Then the relation R is:

Reflexive.
Symmetric.
Transitive.
None of these.

$\cos[\tan^{-1}\{\sin(\cot^{-1}\text{x})\}]$ is equal to:

$\sqrt{\frac{\text{x}^2+2}{\text{x}^3+3}}$
$\sqrt{\frac{\text{x}^2+2}{\text{x}^2+1}}$
$\sqrt{\frac{\text{x}^2+1}{\text{x}^2+2}}$
None of these.

Which of the following is a homogeneous differnetial equation?

If for the matrix $A, A^3 = I,$ than $A^{-1} =$

If a line has the direction ratio 18, 12, 4, then its direction cosines are:

$\frac{9}{11},\frac{6}{11},\frac{2}{11}$
$\frac{9}{13},\frac{6}{13},\frac{2}{13}$
$\frac{9}{7},\frac{6}{7},\frac{2}{7}$
None of these