The integral ∫ d x x 2 x 4 + 1 3 4 equals to

I = ∫ d x x 2 x 4 + 1 3 4 = ∫ d x x 5 1 + 1 x 4 3 4 Put, 1 + 1 x 4 = t ⇒ - 4 x 5 d x = d t ∴ I = 1 4 ∫ - d t t 3 4 = - 1 4 × ∫ t - 3 4 dt = - 1 4 t 1 4 1 4 + c = - 1 + 1 x 4 1 4 + c , where c is the constant of integration.

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Mathematics - Indefinite Integration Question with Solution | TestHub

MathematicsIndefinite IntegrationSubstitutionMedium2 minPYQ_2015

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The integral $\int \frac{d x}{x^{2} {(x^{4} + 1)}^{\frac{3}{4}}}$ equals to

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Answer:

Solution:

$I = \int \frac{d x}{x^{2} {(x^{4} + 1)}^{\frac{3}{4}}}$

$= \int \frac{d x}{x^{5} {(1 + \frac{1}{x^{4}})}^{\frac{3}{4}}}$

Put, $1 + \frac{1}{x^{4}} = t$ $\Rightarrow \frac{- 4}{x^{5}} d x = d t$

$∴ I = \frac{1}{4} \int \frac{- d t}{t^{\frac{3}{4}}}$

$= - \frac{1}{4} \times \int t^{\frac{- 3}{4}} dt$ $= \frac{- 1}{4} (\frac{t^{\frac{1}{4}}}{\frac{1}{4}}) + c$

$= - {(1 + \frac{1}{x^{4}})}^{\frac{1}{4}} + c$ , where $c$ is the constant of integration.

Stream:JEESubject:MathematicsTopic:Indefinite IntegrationSubtopic:Substitution

⏱ 2mℹ️ Source: PYQ_2015

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