Evaluate the following integrals: ^1_0 1-x 1+x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^1_0\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{ dx}$

✓

Answer

Let $\text{x}=\cos2\theta$
Differentiating w.r.t. x, we get
$\text{dx}=-2\sin2\theta\text{ d}\theta$
Now, $\text{x}=0\Rightarrow\theta=\frac{\pi}{4}$
$\text{x}=1\Rightarrow\theta=0$
$\therefore\ \int^\limits1_0\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{ dx}=\int^0\limits_\frac{\pi}{4}\sqrt{\frac{1-\cos2\theta}{1+\cos2\theta}}\big(-2\sin2\theta\big)\text{d}\theta$
$=\int_0\limits^\frac{\pi}{4}\sqrt{\frac{1-\cos2\theta}{1+\cos2\theta}}\big(2\sin2\theta\big)\text{d}\theta$ $\Big[\because\sin2\theta=2\sin\theta\cos\theta;\text{ and }\sin^2\theta=\frac{1-\cos2\theta}{2}\Big]$
$=2=\int_0\limits^\frac{\pi}{4}\frac{\sin\theta}{\cos\theta}\cdot\sin2\theta\text{ d}\theta$
$=4\int_0\limits^\frac{\pi}{4}\sin^2\theta\text{ d}\theta$
$=2\int_0\limits^\frac{\pi}{4}\big(1-\cos2\theta\big)\text{d}\theta$
$=2\Big[\theta-\frac{\sin^2\theta}{2}\Big]^{\frac{\pi}{4}}_0$
$=2\Big[\frac{\pi}{4}-\frac{1}{2}\Big]$
$=\frac{\pi}{2}-1$

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 "4 Marks" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$

If $\text{y}=\log\frac{\text{x}^2+\text{x}+1}{\text{x}^2-\text{x}+1}+\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{\sqrt{3}\text{x}}{1-\text{x}^2}\Big),$ find $\frac{\text{dy}}{\text{dx}}$

Find the intervals in which the following functions are increasing or decreasing.
f(x) = x³ - 6x² + 9x + 15

The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

If x = a cos³ $\theta$and y = a sin³ $\theta$, then find the value of $\frac{\text{f}^{2}\text{y}}{\text{dx}}\text{at}\theta = \frac{\pi}{6}.$

The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 year, and the present population is 100000, when will the city have a population of 500000?

Find the equation of the line passing through the points (-1, 2, 1) and parallel to the line $\frac{2\text{x}-1}{4}=\frac{3\text{y}+5}{4}=\frac{2-\text{z}}{3}.$

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\sin\text{x}\cos^4\text{x}\text{ dx}$

Solve the following linear programming problem by graphical method.Under the constraints, maximise $Z=60 x+40 y$.
$
\begin{aligned}
x+2 y & \leq 12 \\
2 x+y & \leq 12 \\
x+\frac{5}{4} y & \geq 5 ; x \geq 0, y \geq 0
\end{aligned}
$

If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}3$ with $\hat{\text{i}}$, $\frac{\pi}4$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, then find $\theta$ and hence, the components of $\vec{\text{a}}$.