Question

intgreation from π/2- -π/2tan x dx
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Answer 1

For an odd function $f(x),$

$\displaystyle\longrightarrow f(-x)=-f(x)\quad\quad\dots(1)$

We know the definite integral property,

$\longrightarrow\displaystyle\int\limits_b^af(x)\ dx=\int\limits_b^af(a+b-x)\ dx$

If $a+b=0,$

$\longrightarrow\displaystyle\int\limits_{-a}^af(x)\ dx=\int\limits_{-a}^af(-x)\ dx$

From (1),

$\longrightarrow\displaystyle\int\limits_{-a}^af(x)\ dx=\int\limits_{-a}^a-f(x)\ dx$

$\longrightarrow\displaystyle\int\limits_{-a}^af(x)\ dx=-\int\limits_{-a}^af(x)\ dx$

$\longrightarrow\displaystyle\int\limits_{-a}^af(x)\ dx+\int\limits_{-a}^af(x)\ dx=0$

$\longrightarrow\displaystyle\int\limits_{-a}^a[f(x)\ dx+f(x)]\ dx=0$

$\longrightarrow\displaystyle\int\limits_{-a}^a2f(x)\ dx=0$

$\longrightarrow\displaystyle2\int\limits_{-a}^af(x)\ dx=0$

$\longrightarrow\displaystyle\underline{\underline{\int\limits_{-a}^af(x)\ dx=0}}$

Since $f(x)=\tan x$ is an odd function,

$\longrightarrow\displaystyle\underline{\underline{\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\tan x\ dx=0}}$