Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\sqrt{2\text{x}-\text{x}^2}\text{dx}$
✓

Answer

Let $\text{I}=\int\sqrt{2\text{x}-\text{x}^2}\text{dx}$
$=\int\sqrt{\text{x}(2-\text{x})}\text{dx}$
Let $\text{x}=1+\sin\text{u}$
or, $\text{dx}=\cos\text{u du}$
$\Rightarrow\text{I}=\int\sqrt{(1+\sin\text{u})(1-\sin\text{u})}\cos\text{u du}$
$\Rightarrow\text{I}=\int\cos^2\text{u du}$
$\Rightarrow\text{I}=\frac{1}{2}\int(\cos2\text{u}+1)\text{du}$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{1}{2}\sin2\text{u}+\text{u}\Big)+\text{C}$
$\Rightarrow\text{I}=\frac{1}{2}(\sin\text{u}\cos\text{u}+\text{u})+\text{C}$
$\Rightarrow\text{I}=\frac{1}{2}\big(\sin\text{u}\sqrt{1-\sin^2\text{u}}+\text{u}\big)+\text{C}$
$\therefore\ \text{I}=\frac{1}{2}(\text{x}-1)\sqrt{2\text{x}-\text{x}^2}+\frac{1}{2}\sin^{-1}(\text{x}-1)+\text{C}$ $\big[\because\text{u}=\sin^{-1}(\text{x}-1)\big]$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Show that the altitude of the right circular cone of maximum volume that can be inscribed

in a sphere of radius $r$ is $\frac{4 r}{3}$.

→
Evaluate the following :

$\tan ^{-1}(1)+\cos ^{-1}\left(\frac{1}{2}\right)+\sin ^{-1}\left(\frac{1}{2}\right)$

→
Evaluate the following determinant:
$\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$
→
Find two numbers whose sum is 15 and when the square of one multiplied by the cube of the other is maximum.
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,$f(x) = -(x - 1)^3(x + 1), x < 1$
→
Write the minimum value of$ f(x) = x^x .$
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}\text{ on }\Big[0,\frac{\pi}{6}\Big]$
→
Evaluate the following integrals:
$\int\cot^5\text{x}\text{ dx}$
→
A kite is $120\ m$ high and $130\ m$ of string is out. If the kite is moving away horizontally at the rate of $52m/ sec$, find the rate at which the string is being paid out.
→
If $2 y=\sqrt{x+1}+\sqrt{x-1}$, show that $4\left(x^2-1\right) y_2+4 x y_1-y=0$.
→