The value of int_{0}^{pi /2} frac{sin^{2/3} x | vedclass

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

Question

(A) Let $I = \int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$  $(1)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-

Vedclass · Accepted Answer

$\pi /4$

Vedclass · Answer

(A) Let $I = \int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$  $(1)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$,we get:$I = \int_{0}^{\pi /2} \frac{\sin^{2/3}(\pi /2 - x)}{\sin^{2/3}(\pi /2 - x) + \cos^{2/3}(\pi /2 - x)} dx$$I = \int_{0}^{\pi /2} \frac{\cos^{2/3} x}{\cos^{2/3} x + \sin^{2/3} x} dx$  $(2)$Adding equations $(1)$ and $(2)$:$2I = \int_{0}^{\pi /2} \frac{\sin^{2/3} x + \cos^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$$2I = \int_{0}^{\pi /2} 1 dx$$2I = [x]_{0}^{\pi /2} = \pi /2$$I = \pi /4$General property: $\int_{0}^{\pi /2} \frac{\sin^{n} x}{\sin^{n} x + \cos^{n} x} dx = \frac{\pi}{4}$.

The value of $\int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$ is

Similar Questions

The value of the integral $\int_{-1}^{1}\left\{\frac{x^{2015}}{e^{\mid x \mid}\left(x^{2}+\cos x\right)}+\frac{1}{e^{\mid{x} \mid}}\right\} d x$ is equal to

The value of $\int_0^{\frac{\pi}{2}} \frac{dx}{1+\tan^3 x}$ is:

If $\int_{0}^{\pi/2} \tan^{n}(x) dx = k \int_{0}^{\pi/2} \cot^{n}(x) dx$,then

Let $f(x) = \begin{cases} -2, & -2 \leq x \leq 0 \\ x-2, & 0 < x \leq 2 \end{cases}$ and $h(x) = f(|x|) + |f(x)|$. Then $\int_{-2}^2 h(x) dx$ is equal to:

If $[ \cdot ]$ represents the greatest integer function,then $\int_{-1}^1 (x[1+\sin(\pi x)]+1) dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module