MCQ
$\int_{}^{} {\sqrt {1 + \sin x} \;dx = } $
- A$\frac{1}{2}\left( {\sin \frac{x}{2} + \cos \frac{x}{2}} \right) + c$
- B$\frac{1}{2}\left( {\sin \frac{x}{2} - \cos \frac{x}{2}} \right) + c$
- C$2\sqrt {1 + \sin x} + c$
- ✓$ - 2\sqrt {1 - \sin x} + c$
✓
Answer
Correct option: D.
$ - 2\sqrt {1 - \sin x} + c$
d
(d)$\int_{}^{} {\sqrt {1 + \sin x} \,dx} = \int_{}^{} {\sqrt {{{\left( {\sin \frac{x}{2} + \cos \frac{x}{2}} \right)}^2}} } dx$
$ = \int_{}^{} {\sin \frac{x}{2}\,dx} + \int_{}^{} {\cos \frac{x}{2}\,dx} = - 2\cos \frac{x}{2} + 2\sin \frac{x}{2} + c$
$ = - 2\left( {\cos \frac{x}{2} - \sin \frac{x}{2}} \right) + c = - 2\sqrt {(1 - \sin x)} + c.$
(d)$\int_{}^{} {\sqrt {1 + \sin x} \,dx} = \int_{}^{} {\sqrt {{{\left( {\sin \frac{x}{2} + \cos \frac{x}{2}} \right)}^2}} } dx$
$ = \int_{}^{} {\sin \frac{x}{2}\,dx} + \int_{}^{} {\cos \frac{x}{2}\,dx} = - 2\cos \frac{x}{2} + 2\sin \frac{x}{2} + c$
$ = - 2\left( {\cos \frac{x}{2} - \sin \frac{x}{2}} \right) + c = - 2\sqrt {(1 - \sin x)} + c.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
$2{x^3} - 6x + 5$ is an increasing function if
→If $\frac{d}{d x}(f(x))=\log x$, then $f(x)$ equals:
→For real numbers $x$ and $y$, we write $ xRy \in $ $x - y + \sqrt 2 $ is an irrational number. Then the relation $R$ is
→If $A, B$ are two $n \times n$ non $-$ singular matrices, then
→The area of the region enclosed by the curve $y=x^3$ and its tangent at the point $(-1,-1)$ is
→The sine of the angle between the straight line $\frac{\text{x}-2}{3}=\frac{\text{y}-3}{4}=\frac{\text{z}-4}{5}$ and the plane 2x - 2y + z = 5 is:
→Choose the correct answer from the given four options.The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is:
A person travels 12km in the southward direction and then travels 5km to the right and then travels 15km toward the right and finally travels 5km towards the east, how far is he from his starting place?
The equation of the plane containing the lines $r = {a_1} + \lambda {a_2}$ and $r = {a_2} + \lambda {a_1}$ is
→A hall has a square floor of dimension $10\, \mathrm{~m} \times 10\, \mathrm{~m}$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $\mathrm{AG}$ and $\mathrm{BH}$ is $\cos ^{-1} \frac{1}{5}$, then the height of the hall (in $meters$) is :
→