Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
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 = C1 + C2]

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 Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Make a rough sketch of the graph of the function y = 4 - x2, 0 < x < 2  and determine the area enclosed by the curve, the x-axis and the lines x = 0 and x = 2
→
If the interest is compounded continuously at 6% per annum, how much worth Rs 100 will be after 10 years? How long will it take to double Rs 1000?
→
Find the cartesian form of the equations of the following planes.
$\vec{\text{r}}=(1+\text{s}+\text{t})\hat{\text{i}}+(2-\text{s}+\text{t})\hat{\text{j}}+(3-2\text{s}+2\text{t})\hat{\text{k}}$ 
→
Evaluate definite integral $\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x$.
→
Evaluate the following integrals:
$\int\limits^{\pi}_0\frac{\text{x}\tan\text{x}}{\sec\text{x}\text{ cosecx}}\text{ dx}$
→
Evaluate: $\int\limits_0^{\pi/2}\frac{\text{x sin x cos x}}{\text{sin}^{4}\text{x + cos}^{4}\text{x}}\text{dx}$.
→
Find the shortest distance between the lines $\frac{\text{x}-2}{-1}=\frac{\text{y}-5}{2}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}-0}{2}=\frac{\text{y}+5}{-1}=\frac{\text{z}-1}{2}.$
→
Verify Rolle's theorem for the following function on the indicated intervals

$\text{f}(\text{x})=\cos2\Big(\text{x}-\frac{\pi}{4}\Big)\text{ on }\Big[0,\frac{\pi}{2}\Big]$

→
Evaluate the following intregals:
$\int\frac{\cos\text{x}}{(1-\sin\text{x})(2-\sin\text{x})}\ \text{dx}$
→
Evaluate the following intregals:
$\int\frac{\text{x}+1}{\sqrt{\text{x}^2+1}}\text{dx}$
→