Question

If f(x) = ∫ [(5x⁸ + 7x⁶)/(x² + 1 + 2x⁷)²] dx, (x ≥ 0) and f(0) = 0, then the value of f(1) is:
(A) -1/2
(B) 1/2

(C) -1/4
(D) 1/4

Answer 1

The value of f(1)is (d)1/4.

1.f(x) = ∫ [(5x⁸ + 7x⁶)/(x² + 1 + 2x⁷)²] dx

=∫5(x⁸/x^14)+7(x^6/x^14)/(x^2/x^7)+(1/x^7)+(2x^7/x^7)dx

[dividing both numerator and denominator by x^14]

= ∫(5x^-6 +7x^-8)/(x^-5+x^-7+2)²

2.now let x^-5+x^-7+2 =t

=(-5x^-6 -7x^-8)dx=dt

= (5x^-6 +7x^-8)dx = -dt

=f(x)=∫-dt/t²= -∫t^-2 dt

=1/t +c

=1/x^-5+x^-7+2 +c=(x^7/2x^7+x²+1) +c

therefore

f(0)=0

therefore c=0

f(1)=1/(2(1)^7+1²+1)

=1/4