int frac{operatorname{cosec}^2 x-2022}{cos ^{ | vedclass

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(B) ધારો કે $I = \int \frac{\operatorname{cosec}^2 x - 2022}{\cos^{2022} x} dx$.આપણે સંકલનને $I = \int \cos^{-2022} x \operatorname{cosec}^2 x dx - 20

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$-2^{1011}$

Vedclass · Answer

(B) ધારો કે $I = \int \frac{\operatorname{cosec}^2 x - 2022}{\cos^{2022} x} dx$.આપણે સંકલનને $I = \int \cos^{-2022} x \operatorname{cosec}^2 x dx - 2022 \int \sec^{2022} x dx$ તરીકે લખી શકીએ.પ્રથમ પદ પર ખંડશઃ સંકલનનો ઉપયોગ કરતા,$u = \cos^{-2022} x$ અને $dv = \operatorname{cosec}^2 x dx$ લો.તેથી $du = -2022 \cos^{-2023} x (-\sin x) dx = 2022 \cos^{-2023} x \sin x dx$ અને $v = -\cot x$ મળે.સૂત્ર $\int u dv = uv - \int v du$ લાગુ પાડતા:$I = \cos^{-2022} x (-\cot x) - \int (-\cot x) (2022 \cos^{-2023} x \sin x) dx - 2022 \int \sec^{2022} x dx$.કારણ કે $\cot x \sin x = \cos x$,સંકલન આ મુજબ થાય:$I = -\frac{\cot x}{\cos^{2022} x} + 2022 \int \frac{\cos x}{\cos^{2023} x} dx - 2022 \int \sec^{2022} x dx$.$I = -\frac{\cot x}{\cos^{2022} x} + 2022 \int \sec^{2022} x dx - 2022 \int \sec^{2022} x dx + C$.$I = -\frac{\cot x}{\cos^{2022} x} + C$.આમ,$f(x) = -\frac{\cot x}{\cos^{2022} x}$.$x = \pi/4$ માટે કિંમત શોધતા: $f(\pi/4) = -\frac{\cot(\pi/4)}{\cos^{2022}(\pi/4)} = -\frac{1}{(1/\sqrt{2})^{2022}} = -\frac{1}{(1/2)^{1011}} = -2^{1011}$.

$\int \frac{\operatorname{cosec}^2 x-2022}{\cos ^{2022} x} d x=f(x)+C \Rightarrow f(\pi / 4)=$

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