Evaluate the following intregals: 1 3 + dx - Vidyadip

Question

Evaluate the following intregals: $\int\frac{1}{\sqrt{3}\sin\text{x}+\cos\text{x}}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{\sqrt{3}\sin\text{x}+\cos\text{x}}\ \text{dx}$
Let $\sqrt{3}=\text{r}\cos\theta,\text{and }1=\text{r}\sin\theta$
$\tan\theta=\frac{1}{\sqrt{3}}$
$\theta=\frac{\pi}{6}$
$\text{r}=\sqrt{3+1}=2$
$\text{I}=\int\frac{1}{\text{r}\cos\theta\sin\text{x}+\text{r}\sin\theta\cos\text{x}}\ \text{dx}$
$=\frac{1}{\text{r}}\int\frac{1}{\sin(\text{x}+\theta)}\text{dx}$
$=\frac{1}{2}\int\text{cosec}(\text{x}+\theta)\text{dx}$
$=\frac{1}{2}\log\Big|\tan\Big(\frac{\text{x}}{2}+\frac{\theta}{2}\Big)\Big|+\text{c}$
$\text{I}=\frac{1}{2}\log\Big|\tan\Big(\frac{\text{x}}{2}+\frac{\pi}{12}\Big)\Big|+\text{C}$

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 "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

Show that $\text{y}=\text{A}\cos\text{x}+\text{B}\sin\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$

Find the mean, variance and standard deviation of the number of tails in three tosses of a coin.

show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is a solution, is $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$

Find the binomial distribution whose mean is 5 and variance $\frac{10}{3}.$

If $y = Ae^{mx} + Be^{nx}$, show that $\frac{{{d^2}y}}{{d{x^2}}} - \left( {m + n} \right)\frac{{dy}}{{dx}} + mny = 0$

Prove that $\int\limits_{0}^{\text{a}}\text{f(x)}\text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{a}-\text{x})\text{dx},$ hence evaluate $\int\limits_{0}^\pi\frac{\text{x}\sin\text{x}}{1+\cos^2\text{x}}\text{dx}.$

If $\text{y}\sin(\text{x}^\text{x}),$ prove that $\frac{\text{dy}}{\text{dx}}=\cos(\text{x}^\text{x})\times\text{x}^\text{x}(1+\log\text{x})$

For what value of x the matrix A is singular?
$\text{A}=\begin{vmatrix}1+\text{x}&7\\3-\text{x}&8 \end{vmatrix}$

Find the value of k in this question, so that the function f is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\sqrt{1+\text{kx}}-\sqrt{1-\text{kx}}}{\text{x}},&\text{if}-1\leq0\\\frac{2\text{x}+1}{\text{x}-1},&\text{if }0\leq\text{x}\leq1\end{cases}$ at x = 0.

Show that $\text{f}(\text{x})=\sin\text{x}-\cos\text{x}$ is an increasing function on $\Big(-\frac{\pi}{4},\frac{\pi}{4}\Big).$