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

can easily be factorised, think about it

differentiate integral g(t) and use L’hopital in f(x)