intlimits_0^{frac{pi }{2}} frac{dx}{cos^6 x + | vedclass

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

Question

(B) Let $I = \int\limits_0^{\frac{\pi }{2}} \frac{dx}{\cos^6 x + \sin^6 x}$.Using the identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$,we have $\cos^6 x +

Vedclass · Accepted Answer

$\pi$

Vedclass · Answer

(B) Let $I = \int\limits_0^{\frac{\pi }{2}} \frac{dx}{\cos^6 x + \sin^6 x}$.Using the identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$,we have $\cos^6 x + \sin^6 x = (\cos^2 x + \sin^2 x)(\cos^4 x - \cos^2 x \sin^2 x + \sin^4 x) = 1 - 3\sin^2 x \cos^2 x$.So,$I = \int\limits_0^{\frac{\pi }{2}} \frac{dx}{1 - 3\sin^2 x \cos^2 x} = \int\limits_0^{\frac{\pi }{2}} \frac{dx}{1 - \frac{3}{4}\sin^2 2x} = \int\limits_0^{\frac{\pi }{2}} \frac{4 dx}{4 - 3\sin^2 2x}$.Let $2x = t$,then $2 dx = dt$,so $dx = \frac{dt}{2}$. When $x=0, t=0$ and when $x=\frac{\pi}{2}, t=\pi$.$I = \int\limits_0^{\pi} \frac{4 \cdot \frac{dt}{2}}{4 - 3\sin^2 t} = 2 \int\limits_0^{\pi} \frac{dt}{4 - 3\sin^2 t} = 4 \int\limits_0^{\frac{\pi}{2}} \frac{dt}{4 - 3\sin^2 t}$.Divide numerator and denominator by $\cos^2 t$: $I = 4 \int\limits_0^{\frac{\pi}{2}} \frac{\sec^2 t dt}{4\sec^2 t - 3	an^2 t} = 4 \int\limits_0^{\frac{\pi}{2}} \frac{\sec^2 t dt}{4(1+	an^2 t) - 3	an^2 t} = 4 \int\limits_0^{\frac{\pi}{2}} \frac{\sec^2 t dt}{4 + 	an^2 t}$.Let $	an t = u$,then $\sec^2 t dt = du$. When $t=0, u=0$ and when $t 	o \frac{\pi}{2}, u 	o \infty$.$I = 4 \int\limits_0^{\infty} \frac{du}{2^2 + u^2} = 4 \left[ \frac{1}{2} 	an^{-1} \left( \frac{u}{2} ight) ight]_0^{\infty} = 2 \left( \frac{\pi}{2} - 0 ight) = \pi$.

$\int\limits_0^{\frac{\pi }{2}} \frac{dx}{\cos^6 x + \sin^6 x}$ is equal to:

Similar Questions

Let $(2^{1-a} + 2^{1+a})$,$f(a)$,$(3^a + 3^{-a})$ be in $A$.$P$. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \frac{dx}{(e^{2x} - e^{-2x})}$ is:

If $\int_{0}^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta = 1 - \frac{1}{\sqrt{2}}$,$(k > 0)$,then the value of $k$ is

Let $f_n = \int_0^{\frac{\pi}{2}} \left(\sum_{k=1}^n \sin^{k-1} x\right) \left(\sum_{k=1}^n (2k-1) \sin^{k-1} x\right) \cos x \, dx$,where $n \in N$. Then $f_{21} - f_{20}$ is equal to $...........$.

Suppose that $f$ and $g$ are integrable on $[a, b]$,then $f+g$ is integrable on ......... .

What is the value of $\int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$? (Here $[t]$ denotes the greatest integer function of the real number $t$.)

Vedclass Test Series

Exam Paper Generator

Online Exam Module