Question

Integrate this please

Answer 1

let x+π/4 =t

differentiate both sides wrt to x

therefore dx=dt

now integration simply reduces to sec(t)dt

we know that
differentiation of secx + tanx = secx(secx+tanx)

thus multiplying and dividing the integration by sect + tant

we get the intergation as
$\frac{sect(sect \: + tant)dt}{(sect \: + tant)}$

let the denominator be k

k = sect+tant

then

dk =sect(sect+tant)dt

thus by substitution, the integration becomes

$\frac{dk}{k}$

which equals

$ln(k)$

which equals

ln(sect + tant)

put the value of t = x+π/4 to get your answer