int sqrt{sin x} cos x , dx = frac{2}{3}(sin x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given the integral is $I = \int \sqrt{\sin x} \cos x \, dx$. Let $t = \sin x$,then $dt = \cos x \, dx$. The integral becomes $I = \int \sqrt{t} \,

Vedclass · Accepted Answer

$(2n\pi, (2n+1)\pi), n \in \mathbb{Z}$

Vedclass · Answer

(C) Given the integral is $I = \int \sqrt{\sin x} \cos x \, dx$. Let $t = \sin x$,then $dt = \cos x \, dx$. The integral becomes $I = \int \sqrt{t} \, dt = \frac{2}{3} t^{3/2} + C = \frac{2}{3} (\sin x)^{3/2} + C$. For the expression $\sqrt{\sin x}$ to be defined in the real number system,we must have $\sin x \ge 0$. Since the expression is in the denominator or part of a derivative chain where $\sin x$ must be strictly positive for the square root function to be differentiable,we consider $\sin x > 0$. The sine function $\sin x$ is positive in the first and second quadrants,which corresponds to the interval $(2n\pi, (2n+1)\pi)$ for any integer $n$.

$\int \sqrt{\sin x} \cos x \, dx = \frac{2}{3}(\sin x)^{3/2} + C$ is valid when $x$ lies in the interval

Similar Questions

$\int \frac{x}{1+x^4} \, dx =$

The value of the integral $\int \frac{\sin(\ln(2 + 2x))}{x + 1} dx$ is

$\int \frac{1}{\sqrt{1 - e^{2x}}} \, dx = $

$\int \sin^3 x \cdot \cos x \, dx = $

$\int \sec^{2/3} x \csc^{4/3} x \, dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module