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Q. If 3015x31+x2+(1+x2)3dx=α2+β3, where α,β are integers, then α+β is equal to

JEE MainJEE Main 2022Integrals

Solution:

Put 1+x2=t2
2dx=2tdt
Xdx=tdt
2115(t21)tdtt2+t3
1521t(t21)t1+tdt
Put 1+t=u2
dt=2udu
1532(u21)21u×2udu
3032(u42u2)du
30(u552u33)32
30[15(3525)23(3323)]
30[15(9342)23(3322)]
30[15×3+8152]
63+162=α2+β3
α=16,β=6
α+β=10