Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}^2+1)\sqrt{\text{x}}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\frac{\text{x}}{(\text{x}^2+1)\sqrt{\text{x}}}\text{ dx}$
Let $\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt}$
$\therefore\ 2\int\frac{\text{t dt}}{(\text{t}^2+1)\text{t}}$
$=2\Big|\frac{\text{dt}}{\text{t}^4+1}\Big|$
Dividing numerator and denominator by $t^2$
$\text{I}=2\int\frac{\frac{\text{t}}{\text{t}^2}}{\big(\text{t}^2+\frac{1}{\text{t}^2}\big)}\text{ dt}$
$=\int\frac{\Big(1+\frac{1}{\text{t}^2}\big)-\Big(1-\frac{1}{\text{t}^2}\Big)}{\Big(\text{t}^2+\frac{1}{\text{t}^2}\Big)}\text{ dt}$
Let $\text{t}-\frac{1}{\text{t}}=\text{z}$
$\Big(1+\frac{1}{\text{t}^2}\Big)\text{ dt}=\text{dz}$ [For $I^{st}$ part]
and, $\text{t}+\frac{1}{\text{t}}=\text{y}$
$\Big(1+\frac{1}{\text{t}^2}\Big)\text{ dt}=\text{dy}$ [For $II^{nd}$​​​​​​​ part]
$\therefore\ \text{I}=\int\frac{\text{dz}}{\text{z}^2+2}-\int\frac{\text{dy}}{\text{y}^2-2}$
$=\frac{1}{\sqrt{2}}\tan^{-1}\Big(\frac{\text{z}}{\sqrt{2}}\Big)-\frac{1}{2\sqrt{2}}\log\bigg|\frac{\text{y}-\sqrt{2}}{\text{y}+\sqrt{2}}\bigg|+\text{C}$
$=\frac{1}{\sqrt{2}}\tan^{1}\Big(\frac{\text{t}^2-1}{\sqrt{2}\text{t}}\Big)-\frac{1}{2\sqrt{2}}\log\bigg|\frac{\text{x}+1-\sqrt{2\text{x}}}{\text{x}+1+\sqrt{2\text{x}}}\bigg|+\text{C}$

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 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

Differentiate $\sin(x^{2} + 1)$with respect to $ x$from first principle.
→
If $\text{y}^\text{x}=\text{e}^{\text{x}-\text{e}},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{(1+\log\text{y})^2}{\log\text{y}}$
→
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
→
Evaluate the following integrals:
$\int(4\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
→
Differentiate the following functions with respect to x:
$(\log\text{x})^\text{x}$
→
Find the equation of a curve passing through the point (0, 0) and whose differential equation is $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}\sin\text{x.}$
→
Find the equatoion of the passing through the points (1, -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = 9.
→
An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.
→
Find the angle between the pairs of lines with direction ratios proportional to
$2, 2, -2$ and $4, 1, 8$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2-1}}{\text{ax}}\Big),\text{x}\neq0$
→