1-e-tan-x-t-1-t-2-dt-1-e-cot-x-1-t-1-t-2-dt-

Question Number 27294 by iy last updated on 04/Jan/18

\underset{1 / e}{\int^{\tan x}} \frac{t}{1 + t^{2}} d t + \underset{1 / e}{\int^{\cot x}} \frac{1}{t (1 + t^{2})} d t =

Commented by prakash jain last updated on 05/Jan/18

(A)

\int_{1 / e}^{\tan x} \frac{t}{1 + t^{2}} d t = \frac{1}{2} [\ln (1 + \tan^{2} x) - \ln (1 + \frac{1}{e^{2}})]

= \frac{1}{2} [\ln (\sec^{2} x) - \ln (1 + \frac{1}{e^{2}})]

= \ln (\sec x) - \frac{1}{2} \ln (1 + \frac{1}{e^{2}})

\int_{1 / e}^{\cot x} \frac{1}{t (1 + t^{2})} d t

\int_{1 / e}^{\cot x} (\frac{t}{t^{2}} - \frac{t}{1 + t^{2}}) d t

= {[\ln t - \frac{1}{2} \ln (1 + t^{2})]}_{1 / e}^{\cot x}

(B)

= \ln \cot x - \frac{1}{2} \ln (1 + \cot^{2} x) - \ln \frac{1}{e} + \frac{1}{2} \ln (1 + \frac{1}{e^{2}})

= \ln \cot x - \ln cosec x - \ln \frac{1}{e} + \frac{1}{2} \ln (1 + \frac{1}{e^{2}})

A + B

\ln \sec x + \ln \cot x - \ln cosec x + 1

= \ln (\sec x \cdot \cot x) - \ln cosec x + 1

= \ln cosec x - \ln cosec x + 1

= 1