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Evaluate : int (0)^(pi//2) 1/(1 + cos x...

Evaluate : ` int _(0)^(pi//2) 1/(1 + cos x) dx `

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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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