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Indefinite Integrals question

2019 · 9 Jan · Shift 2 · Q32
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Indefinite Integrals question

2019 · 9 Jan · Shift 2 · Q32

JEE MainMathematicsIndefinite IntegralsMCQ+4 / −1
If f(x)=∫5x8+7x6(x2+1+2x7)2 dx, (x≥0),f(0)=0,f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),f\left( 0 \right) = 0,f(x)=∫(x2+1+2x7)25x8+7x6​dx,(x≥0),f(0)=0, then the value of f(1)f(1)f(1) is :
  1. A
    −12- {1 \over 2}−21​
  2. B
    −14- {1 \over 4}−41​
  3. C
    12{1 \over 2}21​
  4. D
    14{1 \over 4}41​
View written solutionFree

Correct answer: D

  1. We need to evaluate f(x)=∫5x8+7x6(x2+1+2x7)2 dx,f(x)=\int \frac{5x^8+7x^6}{(x^2+1+2x^7)^2}\,dx,f(x)=∫(x2+1+2x7)25x8+7x6​dx, with the condition f(0)=0f(0)=0f(0)=0, and then find f(1)f(1)f(1).

  2. Observe the denominator: x2+1+2x7=1+x2+2x7.x^2+1+2x^7=1+x^2+2x^7.x2+1+2x7=1+x2+2x7. Let u=1+x2+2x7.u=1+x^2+2x^7.u=1+x2+2x7. Then dudx=2x+14x6=2x(1+7x5).\frac{du}{dx}=2x+14x^6=2x(1+7x^5).dxdu​=2x+14x6=2x(1+7x5). This does not directly match the numerator, so let us inspect the integrand more carefully.

  3. Factor the numerator: 5x8+7x6=x6(5x2+7).5x^8+7x^6=x^6(5x^2+7).5x8+7x6=x6(5x2+7). This suggests trying a substitution involving x7x^7x7 and x2x^2x2 together. Rewrite the denominator as 1+x2+2x7=1+x2(1+2x5).1+x^2+2x^7=1+x^2(1+2x^5).1+x2+2x7=1+x2(1+2x5). A more useful observation is to check whether the integrand is the derivative of a simple rational expression.

Try ddx(x7x2+1+2x7).\frac{d}{dx}\left(\frac{x^7}{x^2+1+2x^7}\right).dxd​(x2+1+2x7x7​). Using the quotient rule:

=\frac{7x^6(x^2+1+2x^7)-x^7(2x+14x^6)}{(x^2+1+2x^7)^2}.$$ Now simplify the numerator: $$7x^6(x^2+1+2x^7)-x^7(2x+14x^6)$$ $$=7x^8+7x^6+14x^{13}-2x^8-14x^{13}$$ $$=5x^8+7x^6.$$ So indeed, $$\frac{5x^8+7x^6}{(x^2+1+2x^7)^2}=\frac{d}{dx}\left(\frac{x^7}{x^2+1+2x^7}\right).$$ 4. Therefore, $$f(x)=\frac{x^7}{x^2+1+2x^7}+C.$$ 5. Use the condition $f(0)=0$: $$f(0)=\frac{0^7}{0^2+1+2\cdot 0^7}+C=0+C=C.$$ Hence, $$C=0.$$ So, $$f(x)=\frac{x^7}{x^2+1+2x^7}.$$ 6. Now compute $f(1)$: $$f(1)=\frac{1^7}{1^2+1+2\cdot 1^7}= rac{1}{1+1+2}= rac{1}{4}.$$ 7. Hence the correct option is $$\boxed{\text{D }\frac14}.$$
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