∫logx2dx=xfx2+Axfx-1+c, then
fx=logx,A=2
fx=logx,A=-2
fx=-logx,A=2
fx=-logx,A=-2
Explanation for the correct option:
Finding the value using the substitution method:
Consider the given equation,
∫logx2dx=xfx2+Axfx-1+c......(1)
∫logx2dx=∫logx2.1.dx
We know that,
∫uvdx=u∫v-∫du∫vdxdx
∫logx2.1.dx=logx2x-∫2logx×1x×xdx∫logx2.1.dx=xlogx2-2xlogx-∫1x×xdx∫logx2.1.dx=xlogx2-2xlogx-x+c∫logx2.1.dx=xlogx2-2xlogx-1+c.......(2)
Comparing equations 1 and 2 we get
xlogx2-2xlogx-1+c=xfx2+Axfx-1+c
Then,
fx=logxA=-2
Hence, the correct answer is Option (B).
Use the factor theorem to determine whether g(x) is a factor of f(x)
f(x)=22x2+5x+2;g(x)=x+2