Evaluate : ( + )dx - Vidyadip

Question

Evaluate :$\int\Big(\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}\Big)\text{dx}$

✓

Answer

$\text{I}=\int\Big(\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}\Big)\text{dx}=\int\frac{\cos\text{x}+\sin\text{x}}{\sqrt{\sin\text{x}\cos\text{x}}}\text{dx}$
Putting sin x – cos x = t, so that (cos x + sin x) dx = dt
and sin x cos x =$\frac1 2(1 – t^2)$
$\therefore\ \text{I}=\sqrt2\int\frac{\text{dt}}{1-\text{t}^2}=\sqrt2\sin^{-1}\text{t}+\text{c}$
$=\sqrt2\sin^{-1}(\sin\text{x}-\cos\text{x})+\text{c}$

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

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^2=4\text{y}\text{ at }(2,1)$

Find the particular solution of the differential equation $\text{(x - y)} \frac{\text{dy}}{\text{dx}} = \text{(x + 2y),}$ given that y = 0 when x = 1.

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1-\cos\theta)\text{at}\theta=-\frac{\pi}{2}$

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.

Given the function $\text{f(x)}=\frac{1}{\text{x}+2}.$ Find the points of discontinuity of the composite function y = f(f(x)).

Differentiate $(\cos\text{x})^{\sin\text{x}}$ with respect to $(\sin\text{x})^{\cos\text{x}}$

Solve the following differential equation:$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}\text{e}^{\text{x}}$

Differentiate the following functions with respect to x:
$(\sin\text{x})^{\cos\text{x}}$

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:
$2\text{x}+4\text{y}\leq8$
$3\text{x}+\text{y}\leq6$
$\text{x}+\text{y}\leq4$
$\text{x}\geq0,\text{y}\geq0$

Find the area of the bounded by the curve xy - 3x - 2y = 0, x-axis and the lins x = 3, x = 4.