Question

definite integrals...question 16)2)

Answer 1

here you can use integration by part

log (1+x).1/(1+x^2)
here
log (1+x) is 1st function
1/(1+x^2) is 2nd function
now
integration log (1+x)/(1+x^2) is

log(1+x) integ 1/(1+x^2)-integ {1/(1+x^2).1/(1+x)
=> log (1+x).tan^-1(x) -integ{1/(1+x^2)(1+x)

for second part use partial fraction.
=>log (1+x).tan^-1 (x)+1/4ln (1+x^2)-1/2tan^-1 (x)-1/2ln (1+x)

now put limit
then answer is ln2.tan^-1 (2)-1/4ln2-pi/8

Answer 2

log (1+x) is 1st function  1/(1+x^2) is 2nd function now  integration log (1+x)/(1+x^2) is   log(1+x) integ 1/(1+x^2)-integ {1/(1+x^2).1/(1+x) => log (1+x).tan^-1(x) -integ{1/(1+x^2)(1+x)

now put limit then answer is ln2.tan^-1 (2)-1/4ln2-pi/8