IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

piit.jak

prove that

_{[ 0]}

^{[a ]} log(1+ax) / 1+ x² dx

is 1/2 ( tan^-1a) log (1+a²)

raulrag009

sboosy

(0 to a)

log(1+ax)/(1+x²) dx

partially differentiating keepingx constant

(0 to a)

1/(1+ax) * x/(1+x²) dx

now for this integral we apply partial fractions technique

numerator is written as x-a+a

with x-a we get

(0 to a)

a/(a+x) dx -(0 to a)

x/(1+x²) dx +a (0 to a)

dx/(1+ax)(1+x²)

now all are easily integrable and further integration is also easy to obtain

1/2 tan^-1alog(1+a²)

piit.jak

raulrag p009 how sis you change the upper limit to tan inverse a.

also you are lacking a 1/2 in your final answer.