Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS5 Marks
Question
$\int\sin^{-1}\sqrt{\frac{\text{x}}{\text{a}+\text{x}}}\text{dx}$
Hint: Put $\text{x} = \text{a}\tan^2θ$
✓

Answer

Let $\text{I}=\int\sin^{-1}\sqrt{\frac{\text{x}}{\text{a}+\text{x}}}\text{dx}$
Put $\text{x}=\text{a}\tan^2\theta$
$\Rightarrow\ \text{dx}=2\text{a}\tan\theta\sec^2\theta\text{ d}\theta$
$\therefore\ \text{I}=\int\sin^{-1}\sqrt{\frac{\text{a}\tan^2\theta}{\text{a}+\text{a}\tan^2\theta}}(2\text{a}\tan\theta\cdot\sec^2\theta)\text{d}\theta$
$=2\text{a}\int\sin^{-1}\Big(\frac{\tan\theta}{\sec\theta}\Big)\tan\theta\cdot\sec^2\theta\text{ d}\theta$
$=2\text{a}\int\sin^{-1}(\sin\theta)\tan\theta\cdot\sec^2\theta\text{ d}\theta$
$=2\text{a}\int\theta\cdot\tan\theta\sec^2\theta\text{ d}\theta\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{I} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ II}$
$=2\text{a}\bigg[\theta\cdot\int\tan\theta\cdot\sec^2\theta\text{ d}\theta-\int\Big(\frac{\text{d}}{\text{d}\theta}\theta\cdot\int\tan\theta\cdot\sec^2\theta\text{ d}\theta\Big)\text{d}\theta\bigg]$
$\begin{bmatrix} \text{let} \tan\theta=\text{t} \\\sec^2\theta\text{d}\theta=\text{dt}\\ \ \ \ \ \ \int\tan\theta\sec^2\theta\text{ d}\theta=\int\text{tdt}=\frac{\text{t}^2}{2}=\frac{\tan^2\theta}{2}\end{bmatrix}$
$=2\text{a}\Big[\theta\cdot\frac{\tan^2\theta}{2}-\int\frac{\tan^2\theta}{2}\text{d}\theta\Big]$
$=\text{a}\theta\tan^2\theta-\text{a}\int(\sec^2\theta-1)\text{d}\theta$
$=\text{a}\theta\cdot\tan^2\theta-\text{a}\tan\theta+\text{a}\theta+\text{C}$
$=\text{a}\bigg[\frac{\text{x}}{\text{a}}\tan^{-1}\sqrt{\frac{\text{x}}{\text{a}}}+\tan^{-1}\sqrt{\frac{\text{x}}{\text{a}}}\bigg]+\text{C}$

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 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

Evaluate: $\int(\text{x} - 3 )\sqrt{\text{x}^{2} + 3 \text{x}- 18 }\text{ dx}.$
→
Differentiate w.r.t. x the function in Exercise:
$(\sin\text{x}-\cos\text{x)}^{(\sin\text{x}-\cos\text{x})},\ \frac{\pi}{4}<\text{x}<\frac{3\pi}{4}$
→
Find the equation of the plane that is perpendicular to the plane $5x + 3y + 6z + 8 = 0$ and which contains the line of intersection of the planes $x + 2y + 3z - 4 = 0, 2x + y - z + 5 = 0.$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=​​\begin{cases}-2,&\text{if }\text{ x}\leq-1\\2\text{x},&\text{if }-1<\text{x}\leq1\\2,&\text{if }\text{ x}>1\end{cases}$
→
Show that the lines $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$ and $\frac{x-2}{2}=\frac{y-1}{3}=\frac{z+1}{-2}$ intersect and find their point of intersection.
→
Prove by Mathematical Induction that $(A′)^n = (A^n)′,$ where $\text{n}\in\text{N}$ for any square matrix $A.$
→
Draw a rough sketch of the region $\{(x, y) : y^2 < 3x, 3x^2 + < 16\}$ and find the area by the region using mwthod of integration.
→
Let $A = R - {3}, B = R - {1}.$ Let $f : A \rightarrow B$ be defined by $\text{f}(\text{x})=\frac{\text{x}-2}{\text{x}-3}\ \forall\ \text{x}\in\text{A}.$ Then show that $f$ is bijective.
→
Evaluate the following definite integrals:
$\int\limits_{\frac{\pi}{2}}^{\pi}\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$
→
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
→