If $f(x)=\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x,(x \geq 0)$ and $f(0)=0$, then the value of $f(1)$ is:

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Indefinite Integrals

f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0)

and

f (0) = 0

, then the value of

f (1)

is:

Answer: (D)

Divide top and bottom by

x^{14}

to expose a substitution:

f (x) = \int \frac{5 x ^{- 6} + 7 x ^{- 8}}{( 2 + x ^{- 5} + x ^{- 7} ) ^{2}} d x

. Let

u = 2 + \frac{1}{x ^{5}} + \frac{1}{x ^{7}}

, so

d u = (- \frac{5}{x ^{6}} - \frac{7}{x ^{8}}) d x

, turning the integral into

- \int \frac{d u}{u ^{2}} = \frac{1}{u} + C = \frac{x ^{7}}{2 x ^{7} + x ^{2} + 1} + C

. The condition

f (0) = 0

forces

C = 0

, so

f (x) = \frac{x ^{7}}{2 x ^{7} + x ^{2} + 1}

and

f (1) = \frac{1}{2 + 1 + 1} = \frac{1}{4}

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option