Integrate the function x+2 4 x-x^ 2 - Vidyadip

Question

Integrate the function $\frac{x+2}{\sqrt{4 x-x^{2}}}$

✓

Answer

Let $x + 2 = A \frac{d}{d x}\left(4 x-x^{2}\right)+B$
$\Rightarrow x + 2 = A(4 -2x) + B$
Now, equating the coefficients of $x$ and constant term on both sides, we get,
$-2A = 1$
$\Rightarrow A=-\frac{1}{2}$
and $4A + B = 2$
$\Rightarrow B = 4$
$\Rightarrow x+2=-\frac{1}{2}(4-2 x)+4$
Now, $\int \frac{x+2}{\sqrt{4 x-x^{2}}} d x=\int \frac{-\frac{1}{2}(4-2 x)+4}{\sqrt{4 x-x^{2}}} d x$
$\Rightarrow -\frac{1}{2} \int \frac{(4-2 x)}{\sqrt{4 x-x^{2}}} d x+4 \int \frac{1}{\sqrt{4 x-x^{2}}} d x$
Now, let us consider, $\int \frac{(4-2 x)}{\sqrt{4 x-x^{2}}} d x$
Let $4x - x^2 = t$
$\Rightarrow (4 - 2x) dx = dt$
$\therefore \int \frac{(4-2 x)}{\sqrt{4 x-x^{2}}} d x=\int \frac{d t}{\sqrt{t}}=2 \sqrt{t}=2 \sqrt{4 x-x^{2}}.......(i)$
And, Now let us consider, $\int \frac{1}{\sqrt{4 x-x^{2}}} d x$
Then, $4x - x^2 = -(-4x + x^2)$
$= (-4x + x^2 + 4 - 4)$
$= 4 - (x - 2)^{2 }$
$= (2)^2 - (x - 2)^{2 }$
$\therefore \int \frac{1}{\sqrt{4 x-x^{2}}} d x=\sin ^{-1}\left(\frac{x-2}{2}\right) ......(ii)$
using eq. $(i)$ and $(ii),$ we get,
$\Rightarrow \int \frac{x+2}{\sqrt{4 x-x^{2}}} d x=-\sqrt{4 x-x^{2}}+4 \sin ^{-1}\left(\frac{x-2}{2}\right)+C$

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

Similar questions

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs. 300 and on one item of B is Rs. 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

Show that the points $\text{A}\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big),\ \text{B}\big(\hat{\text{i}}-3\hat{\text{j}}-5\hat{\text{k}}\big),$ $\text{C}\big(3\hat{\text{i}}-4\hat{\text{j}}-4\hat{\text{k}}\big)$ are the vertices of a right angled triangle.

Solve the following differential equations:$(1+\text{y}^2)\tan^{-1}\text{xdx}+2\text{y}(1+\text{x}^2)\text{dy}=0$

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

Suppose a girl throws a die. If she gets a $5$ or $6,$ she tosses a coin $3$ times and notes the number of heads. If she gets $1, 2, 3$ or $4$ she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw $1, 2, 3$ or $4$ with the die?

Show that the points A, B, C with position vectors $\vec{\text{a}}-2\vec{\text{b}}+3\vec{\text{c}},\ 2\vec{\text{a}}+3\vec{\text{b}}-4\vec{\text{c}}$ and $-7\vec{\text{b}}+10\vec{\text{c}}$ are collinear.

Find the intervals in which $f(x) = (x + 2)e^{-x}$ is increasing or decreasing.

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.
If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

kg per bag
	Brand P	Brand Q
Nitrogen Phosphoric acid Potash Chlorine	3 1 3 1.5	3.5 2 1.5 2

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}=\text{x}^2\cos^2\text{x}$

Evaluate the following integrals as limit of sum:
$\int\limits^3_{2}\big(2\text{x}^2+1\big)\text{ dx}$