Question
Mathematics Question on integral
If ∫(x+2)(x2+1)dx=alog1+x2+btan−1x+51log∣x+2∣+C, then
A
a=10−1,b=5−2
B
a=101,b=5−2
C
a=10−1,b=52
D
a=101,b=52
Answer
a=10−1,b=52
Explanation
Solution
We have, I=∫(x+2)(x2+1)dx Let (x+2)(x2+1)1=x+2A+x2+1Bx+C ⇒1=A(x2+1)+Bx(x+2)+C(x+2)...(i) Put x=0 in (i), we get A+2C=1 Put x=−2 in (i), we get A=51 ⇒C=52 Put x=1 in (i), we get 1=2A+3B+3C, we get B=5−1 ⇒∫(x+2)(x2+1)dx=51∫x+2dx−51∫x2+1xdx+52∫x2+1dx =51log∣x+2∣−101logx2+1+52tan−1x+C Hence, a=10−1 and b=52