જો I=int_0^{pi / 2} frac{d x}{5+3 sin x}=lamb | vedclass

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

Question

(C) આપેલ છે, $I=\int_0^{\pi / 2} \frac{d x}{5+3 \sin x}=\lambda 	an ^{-1}\left(\frac{1}{2}ight)$.આદેશ $\sin x = \frac{2 	an(x/2)}{1+	an^2(x/2)}$

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(C) આપેલ છે, $I=\int_0^{\pi / 2} \frac{d x}{5+3 \sin x}=\lambda 	an ^{-1}\left(\frac{1}{2}ight)$.આદેશ $\sin x = \frac{2 	an(x/2)}{1+	an^2(x/2)}$ નો ઉપયોગ કરતા:$I = \int_0^{\pi / 2} \frac{dx}{5 + 3 \left( \frac{2 	an(x/2)}{1+	an^2(x/2)} ight)} = \int_0^{\pi / 2} \frac{(1+	an^2(x/2)) dx}{5 + 5	an^2(x/2) + 6	an(x/2)} = \int_0^{\pi / 2} \frac{\sec^2(x/2) dx}{5	an^2(x/2) + 6	an(x/2) + 5}$.ધારો કે $t = 	an(x/2)$, તેથી $dt = \frac{1}{2} \sec^2(x/2) dx$, એટલે કે $\sec^2(x/2) dx = 2 dt$.જ્યારે $x=0, t=0$; જ્યારે $x=\pi/2, t=1$.$I = \int_0^1 \frac{2 dt}{5t^2 + 6t + 5} = \frac{2}{5} \int_0^1 \frac{dt}{t^2 + \frac{6}{5}t + 1} = \frac{2}{5} \int_0^1 \frac{dt}{(t + 3/5)^2 + 1 - 9/25} = \frac{2}{5} \int_0^1 \frac{dt}{(t + 3/5)^2 + (4/5)^2}$.સૂત્ર $\int \frac{dx}{x^2+a^2} = \frac{1}{a} 	an^{-1}(\frac{x}{a})$ નો ઉપયોગ કરતા:$I = \frac{2}{5} \cdot \frac{5}{4} \left[ 	an^{-1} \left( \frac{t + 3/5}{4/5} ight) ight]_0^1 = \frac{1}{2} \left[ 	an^{-1} \left( \frac{5t+3}{4} ight) ight]_0^1 = \frac{1}{2} \left( 	an^{-1}(2) - 	an^{-1}(3/4) ight)$.સૂત્ર $	an^{-1} A - 	an^{-1} B = 	an^{-1} \left( \frac{A-B}{1+AB} ight)$ નો ઉપયોગ કરતા:$I = \frac{1}{2} 	an^{-1} \left( \frac{2 - 3/4}{1 + 2(3/4)} ight) = \frac{1}{2} 	an^{-1} \left( \frac{5/4}{1 + 3/2} ight) = \frac{1}{2} 	an^{-1} \left( \frac{5/4}{5/2} ight) = \frac{1}{2} 	an^{-1} \left( \frac{1}{2} ight)$.$\lambda 	an^{-1}(1/2)$ સાથે સરખાવતા, આપણને $\lambda = 1/2$ મળે છે.

જો $I=\int_0^{\pi / 2} \frac{d x}{5+3 \sin x}=\lambda \tan ^{-1}\left(\frac{1}{2}\right)$, તો $\lambda=$

Similar Questions

$\int_0^{16} \frac{\sqrt{x}}{1+\sqrt{x}} d x=$

જો $\frac{d}{dx}\{f(x)\} = g(x)$ હોય, તો $\int_a^b f(x) g(x) dx$ ની કિંમત શું થાય?

નિશ્ચિત સંકલન $\int\limits_0^{\frac{\pi }{2}} {\frac{{\sin 5x}}{{\sin x}}\,dx} $ નું મૂલ્ય છે

સંકલન $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x - x \cos x)}{x(x + \sin x)} dx$ નું મૂલ્ય શું છે?

સંકલન $\int_0^2 \frac{\sqrt{x}(x^2 + x + 1)}{(\sqrt{x}+1)(\sqrt{x^4+x^2+1})} dx$ નું મૂલ્ય કેટલું થાય?

Vedclass Test Series

Exam Paper Generator

Online Exam Module