If f x = ∫ 5 x 8 + 7 x 6 x 2 + 1 + 2 x 7 2 d x , x ≥ 0 , f 0 = 0 and f 1 = 1 K , then the value of K is

f x = ∫ 5 x 8 + 7 x 6 d x x 14 x - 5 + x - 7 + 2 2 Let x - 5 + x - 7 + 2 = t - 5 x - 6 - 7 x - 8 d x = d t ⇒ f x = ∫ - d t t 2 = 1 t + c f x = x 7 x 2 + 1 + 2 x 7 f 1 = 1 4

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Mathematics - Indefinite Integration Question with Solution | TestHub

MathematicsIndefinite IntegrationSubstitutionMedium2 minPYQ_2021

MathematicsMediumnumerical

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

Answer:

4.00

Solution:

$f (x) = \int \frac{(5 x^{8} + 7 x^{6}) d x}{x^{14} {(x^{- 5} + x^{- 7} + 2)}^{2}}$

Let $x^{- 5} + x^{- 7} + 2 = t$

$(- 5 x^{- 6} - 7 x^{- 8}) d x = d t$

$\Rightarrow f (x) = \int - \frac{d t}{t^{2}} = \frac{1}{t} + c$

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

$f (1) = \frac{1}{4}$

Stream:JEESubject:MathematicsTopic:Indefinite IntegrationSubtopic:Substitution

⏱ 2mℹ️ Source: PYQ_2021

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