જો int frac{dx}{5 + 4cos x} = lambda tan^{-1} | vedclass

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Question

(A) આપણે $\cos x = \frac{1 - 	an^2(x/2)}{1 + 	an^2(x/2)}$ આદેશનો ઉપયોગ કરીએ છીએ.$\int \frac{dx}{5 + 4\cos x} = \int \frac{dx}{5 + 4\left( \frac{1 -

Vedclass · Accepted Answer

$\lambda = 2/3$

Vedclass · Answer

(A) આપણે $\cos x = \frac{1 - 	an^2(x/2)}{1 + 	an^2(x/2)}$ આદેશનો ઉપયોગ કરીએ છીએ.$\int \frac{dx}{5 + 4\cos x} = \int \frac{dx}{5 + 4\left( \frac{1 - 	an^2(x/2)}{1 + 	an^2(x/2)} ight)}$$= \int \frac{(1 + 	an^2(x/2)) dx}{5(1 + 	an^2(x/2)) + 4(1 - 	an^2(x/2))}$$= \int \frac{\sec^2(x/2) dx}{9 + 	an^2(x/2)}$ધારો કે $	an(x/2) = y$,તો $\frac{1}{2} \sec^2(x/2) dx = dy$,તેથી $\sec^2(x/2) dx = 2 dy$.$= \int \frac{2 dy}{9 + y^2} = 2 \int \frac{dy}{3^2 + y^2} = 2 \cdot \frac{1}{3} 	an^{-1} \left( \frac{y}{3} ight) + C$$= \frac{2}{3} 	an^{-1} \left( \frac{1}{3} 	an \frac{x}{2} ight) + C$.$\lambda 	an^{-1} (m 	an(x/2))$ સાથે સરખાવતા,આપણને $\lambda = 2/3$ અને $m = 1/3$ મળે છે.

જો $\int \frac{dx}{5 + 4\cos x} = \lambda \tan^{-1} \left( m \tan \frac{x}{2} \right) + C$ હોય,તો:

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