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Evaluate : ` int _(0)^(pi//2) 1/(1 + cos x) dx ` |
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Answer» Given, `underset(0)overset(pi/2) int1/(1+cos x) dx ` ` = underset(0)overset(pi/2) int 1/(2 cos^(2). x/2) dx` ` = 1/2 underset(0)overset(pi/2) intsec^(2) (x/2) dx ` `[" Let " x/2 = u " " :. 1/2 dx = du " when" x gt 0, u gt 0 and x gt pi/2, u gt pi/4 ]` ` :. " " = 1/2 underset(0)overset(pi/2) int sec^(2)(x/2) dx ` = `underset(0)overset(pi/2) intsec^(2) u" " du= [tan u]_(0)^(pi/4)` ` = tan pi/4 - tan 0 = 1` |
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