Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS5 Marks
Question
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\cot\text{x}}}{\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\cot\text{x}}}{\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}}\text{ dx}\ ...(\text{i})$
$=\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\cot\big(\frac{\pi}{2}-\text{x}\big)}}{\sqrt{\cot\big(\frac{\pi}{2}-\text{x}\big)}+\sqrt{\tan\big(\frac{\pi}{2}-\text{x}\big)}}\text{ dx}$ $\Big[\text{Using},\int\limits^{\text{a}}_0\text{f(x)}\text{dx}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})\text{dx}\Big]$
$=\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\tan\text{x}}}{\sqrt{\tan\text{x}}+\sqrt{\cot\text{x}}}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii)
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\bigg(\frac{\sqrt{\cot\text{x}}}{\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}}+\frac{\sqrt{\tan\text{x}}}{\sqrt{\tan\text{x}}+\sqrt{\cot\text{x}}}\bigg)\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\text{dx}$
$=\Big[\text{x}\Big]^{\frac{\pi}{2}}_0=\frac{\pi}{2}$
Hence, $\text{I}=\frac{\pi}{4}$

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

The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1cm/ sec, find the rate of increase of the outer radius when the radii are 4cm and 8cm respectively.
→
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{e}^{\theta}\Big(\theta+\frac{1}{\theta}\Big)\text{ and y}=\text{e}^{-\theta}\Big(\theta-\frac{1}{\theta}\Big)$
→
Reduce the equation $\vec{\text{r}}\cdot(\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}})+6=0$ to the normal form and, hence, find the length of the perpendicular from the origin to the plane.
→
Show that the three lines with direction cosines $\frac{12}{13},\frac{-3}{13},\frac{-4}{13},\frac{4}{13},\frac{12}{13},\frac{3}{13},\frac{3}{13},\frac{-4}{13},\frac{12}{13}$ are mutually perpendicular.
→
Maximise Z = x + y, subject to $\text{x}-\text{y}\leq-1,\ -\text{x}+\text{y}\leq0,\ \text{x},\ \text{y}\geq0.$
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{6}}\cos\text{x }\cos2\text{x}\text{ dx}$
→
A box manufacturer makes large and small boxes from a large piece of cardboard. The large boxes require $4$ sq. metre per box while the small boxes require $3 sq.$ metre per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If $60 sq.$ metre of cardboard is in stock, and if the profits on the large and small boxes are $Rs. 3$ and $Rs. 2$ per box, how many of each should be made in order to maximize the total profit?
→
Using integration, find the area of the triangle $ABC$ coordinatrs of whise vertices $A(4, 1), B(6, 6)$ and $C(8, 4).$
→
If x = a (cos t + t sin t) and y = a (sin t – t cos t), 0 < t < $\frac{\pi}{2},$ $\text{find }\frac{\text{d}^{2}\text{x}}{\text{dt}^{2}},\frac{\text{d}^{2}\text{y}}{\text{dt}^{2}}\text{ and }\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$.
→
Find the area of the region enclosed by the parabola $x^2 = y$ and the line $y = x + 2.$
→