Evaluate the following integrals: x (x-3) x+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}-3)\sqrt{\text{x}+1}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{x}}{(\text{x}-3)\sqrt{\text{x}+1}}\text{ dx}$
$=\int\frac{(\text{x}-3)+3}{(\text{x}-3)\sqrt{\text{x}+1}}\text{ dx}$
$\text{I}=\int\frac{\text{dx}}{\sqrt{\text{x}+1}}+3\int\frac{\text{dx}}{(\text{x}-3)\sqrt{\text{x}+1}}\ ...(\text{i})$
Now, $\int\frac{\text{dx}}{\sqrt{\text{x}+1}}=2\sqrt{\text{x}+1}+\text{C}_1$
and, $\int\frac{\text{dx}}{\sqrt{\text{x}+2}}=2\sqrt{\text{x}+2}+\text{C}_2$
$\int\frac{\text{dx}}{(\text{x}-3)\sqrt{\text{x}+1}}$
Let $\text{x}+1=\text{t}^2$
$\text{dx}=2\text{t dt}$
$\int\frac{\text{dx}}{(\text{x})-3\sqrt{\text{x}+1}}=2\int\frac{\text{t dt}}{(\text{t}^2-4)\text{t}}$
$=2\Big|\frac{\text{dt}}{\text{t}^2-4}\Big|$
$=\frac{2}{2\times2}\log\Big|\frac{\text{t}-2}{\text{t}+2}\Big|+\text{C}_2$
$\therefore\ \int\frac{\text{dx}}{(\text{x}-3)\sqrt{\text{x}+1}}=\frac{1}{2}\log\bigg|\frac{\sqrt{\text{x}+1}-2}{\sqrt{\text{x}}+1+2}\bigg|+\text{C}_2$
Thus, from (i)
$\text{I}=2\sqrt{\text{x}+1}+\frac{3}{2}\log\bigg|\frac{\sqrt{\text{x}+1}-2}{\sqrt{\text{x}+1}+2}\bigg|+\text{C} [$When $C = C_1 + C_2]$

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 Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Two tailors A and B earn Rs 150 and Rs 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.

Evaluate the following integrals:
$\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}(1+\sin\text{x})}{1+\cos^2\text{x}}\text{ dx}$

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

If $\vec{\text{a}}$ be the position vector whose tip is (5, -3), find the coordinates of a point B such that $\overrightarrow{\text{AB}}=\vec{\text{a}}$, the coordinates of A being (4, -1).

$\text{If A} = \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix}, \text{B} = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}, \text{C} = \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix},$ then calculate AC, BC and (A + B) C. Also verify that (A + B) C = AC + BC.

If $y = 3 \cos (\log x) + 4 \sin (\log x).$ Show that $x^2y_2 + xy_1 + y = 0$

Find the value of 'a' for which the function f defined as$ \begin{matrix} & \text{a sin}\frac{\pi}{2}\text{(x + 1)}, & x\leq0 \\ \text{f(x)} \\ & \frac{\text{tan x - sin x}}{\text{x}^{3}}, & x<0 \\ \end{matrix}$
is continuous at X = 0.

Differentiate $\tan^{-1 }\Bigg[\frac{\sqrt{\text{1+x}^{2}-1}}{\text{x}}\Bigg]$ with respect to $x.$

Show that $\text{y}=4\text{ax}$ is a solution of the differential equation $\text{y}=\text{x}\frac{\text{d}\text{y}}{\text{dx}}+\text{a}\frac{\text{dx}}{\text{dy}}.$