Evaluate: _ /6 ^ /3 dx 1 + . - Vidyadip

Question

Evaluate: $\int\limits_{\pi/6}^{\pi/3}\frac{\text{dx}}{1 + \sqrt{\cot\text{x}}}.$

✓

Answer

$\text{I} = \int\limits_{\pi/6}^{\pi/3}\frac{\text{dx}}{1 + \sqrt{\cot\text{x}}} = \int\limits_{\pi/6}^{\pi/3}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}} + \sqrt{\cos\text{x}}}\text{dx}$
$ = \int\limits_{\pi/6}^{\pi/3}\frac{\sqrt{\sin\bigg(\frac{\pi}{3} + \frac{\pi}{6} - \text{x}}\bigg)}{\sqrt{\sin\bigg(\frac{\pi}{3} + \frac{\pi}{6} - \text{x}\bigg) + \sqrt{\cos\bigg(\frac{\pi}{3} + \frac{\pi}{6} - \text{x}}\bigg)}}\text{dx}$
$\therefore\text{I} = \int\limits_{\pi/6}^{\pi/3}\frac{\sqrt{\cos\text{x}}}{\sqrt{\cos\text{x}} +\sqrt{\sin\text{x}}}\text{dx}$
Adding we get, $2 \text{I} = \int\limits_{\pi/6}^{\pi/3}\text{dx} = \big[\text{x}\big]_{\pi/6}^{\pi/3} = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}$
$\therefore\text{I} = \frac{\pi}{12}.$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Prove the following results:
$\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}=\tan^{-1}\frac{63}{16}$

If $\text{A}=\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix},$ and $I$ is the identity matrix of order $3,$ show that $A^3 = pI + qA + rA^2.$

If $\text{y}=\tan^{-1}\text{x},$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ in terms of y alone.

Show that the following system of linear equations is consistent and also find solution: $6x + 4y = 2 , 9x + 6y =3$

Find the coordinates of the foot of the perpendicular from the point (2, 3, 7) to the plane 3x - y - z = 7. Also, find the length of the perpendicular.

Prove that the area the region bounded by $\text{y}=\sqrt{\text{x}}, $and x = 2y + 3 in the first and x-asis.

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\frac{(\text{x}^2-1)^3(2\text{x}-1)}{\sqrt{(\text{x}-3)(4\text{x}-1)}}$

Solve the following systems of linear equations by cramer's rule:

Find the angle between the following pair of lines:
$\frac{\text{-x + 2}}{-2}=\frac{\text{y - 1}}{7}=\frac{\text{z + 3}}{-3}$ and $\frac{\text{x + 2}}{-1}=\frac{\text{2y - 8}}{4}=\frac{\text{z -5 }}{4}$
and check whether the lines are parallel or perpendicular.

Evaluvate the following intregals:
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$