First slide

Evaluation of definite integrals

Question

The value of $\int_{1 / e}^{\tan x} \frac{t}{1 + t^{2}} d t + \int_{1 / e}^{\cot x} \frac{d t}{t (1 + t^{2})}$ is

Moderate

A

$1 / 2$
B

1
C

$π / 4$
D

none of these

Solution

$\frac{1}{t (1 + t^{2})} = \frac{1}{t} - \frac{t}{1 + t^{2}}$ So the required integral is equal to
$\begin{array}{l} \int_{\frac{1}{e}}^{\tan x} \frac{t d x}{1 + t^{2}} + \int_{\frac{1}{e}}^{\cot x} \frac{d t}{t} - \int_{\frac{1}{e}}^{\cot x} \frac{t d t}{1 + t^{2}} \\ = {\frac{1}{2} \log (1 + t^{2})|}_{\frac{1}{e}}^{\tan x} + {\log t|}_{\frac{1}{e}}^{\cot x} - {\frac{1}{2} \log (1 + t^{2})|}_{\frac{1}{e}}^{\cot x} \\ = \log | \sec x | + \log | \cot x | + 1 - \log | cosec x | = 1 \end{array}$

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