Integration

denominator is (t+cos)^2 + sin^2

take e^x = t

after converting the expression in terms of t,

divide both nr and dr by t^2

solve the integral

then solve the 1^inf case

take x as 1st

xe^-x^2 as 2nd

write the integral and then expand

as the expression is indipendent of b, f(0)=f(1)

solve further

understand using area under graph

Use king’s to get sin6x/sin2x

take 2x = theta

solve futher

x must be 1 and y must be -1 in order to get sum as 9pi

seeing the limits, we realsie that we should subc x with 1/t

just direct by parts

just use king’s

divide 0 to pi/2 into two intervals,

0 to pi/4

pi/4 to pi/2

convert to sin cos

then use kings

put sec + cos = t

expand ln(1+x) and solve that integral

then do harkate to get ans

take e^x to LHS

then diff both sides

use king’s

just use king’s then open the bracket and integrate each term