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  • Confused! kindly explain, - If and then the value of is : - Integral Calculus - JEE Main

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

and f(0)=0, then the value of f(1) is :

  • Option 1)

    1/2

  • Option 2)

     

    -1/2

  • Option 3)

     

    1/4

  • Option 4)

     

    -1/4

Answers (1)

best_answer

 

Type of integration by substitution -

\int (f(x))^{n}\cdot f{}'(x)dx

\therefore \frac{\left [ f(x) \right ]^{n+1}}{n+1}+c

- wherein

Let f(x)=t

f{}'(x)dx=dt

 

\int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2}dx = \int \frac{5x^8 + 7x^6}{x^{14}(x^{-5} + x^{-7} + 2)^2}dx \\ = \int \frac{5x^{-6} + 7x^{-8}}{(x^{-5} + x^{-7} + 2)^2}dx

Put

 x^{-7} + x^{-5} +2 = t \\\Rightarrow (-7x^{-8} + (-5)x^{-5})dx = dt

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

As

f(0) = 0 \\\Rightarrow f(1) = \frac{1}{4}


Option 1)

1/2

Option 2)

 

-1/2

Option 3)

 

1/4

Option 4)

 

-1/4

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