The value of intlimits_{frac{-pi}{2}}^{frac{p | vedclass

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

Question

(A) Let $I = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + (2017)^x} \, dx$. Using the property $\int_{a}^{b} f(x) \, dx = \int_{a}

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(A) Let $I = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + (2017)^x} \, dx$. Using the property $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$,we have $I = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2(-x)}{1 + (2017)^{-x}} \, dx$. Since $\sin^2(-x) = \sin^2 x$,we get $I = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + \frac{1}{(2017)^x}} \, dx = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{(2017)^x \sin^2 x}{(2017)^x + 1} \, dx$. Adding the two expressions for $I$: $2I = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 x + (2017)^x \sin^2 x}{1 + (2017)^x} \, dx = \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin^2 x \, dx$. Since $\sin^2 x$ is an even function,$2I = 2 \int\limits_{0}^{\frac{\pi}{2}} \sin^2 x \, dx$. $I = \int\limits_{0}^{\frac{\pi}{2}} \sin^2 x \, dx = \int\limits_{0}^{\frac{\pi}{2}} \frac{1 - \cos 2x}{2} \, dx = \left[ \frac{x}{2} - \frac{\sin 2x}{4} \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{4} - 0 = \frac{\pi}{4}$.

The value of $\int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + (2017)^x} \, dx$ is

Similar Questions

The value of $\int_{-2}^{2}\left|3 x^{2}-3 x-6\right| d x$ is ...... .

If $I_n = \int_0^{\pi / 4} \tan^n x \, dx$,then $I_2+I_4, I_3+I_5, I_4+I_6, \ldots$ are in

Let $f:R \to R$ and $g:R \to R$ be continuous functions,then the value of the integral $\int_{-\pi/2}^{\pi/2} [f(x) + f(-x)][g(x) - g(-x)] \, dx$ is:

If $\int_0^{2 \pi} |x \sin x| \, dx = k \pi$,then $k =$

If the value of the integral $\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$,then a value of $\alpha$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module