dx-1-x-1-4-x-

Question Number 157977 by tounghoungko last updated on 30/Oct/21

\int \frac{d x}{(1 + \sqrt[4]{x}) \sqrt{x}} = ?

Answered by puissant last updated on 30/Oct/21

Ω = \int \frac{d x}{(1 + \sqrt[4]{x}) \sqrt{x}}

u = \sqrt{x} \to d u = \frac{1}{2 \sqrt{x}} d x \to d x = 2 \sqrt{x} d u

\Rightarrow Ω = \int \frac{2 \sqrt{x} d u}{(1 + \sqrt{u}) \sqrt{x}} = 2 \int \frac{d u}{(1 + \sqrt{u})}

t = \sqrt{u} \to t^{2} = u \to 2 t d t = d u

\Rightarrow Ω = 2 \int \frac{2 t d t}{(1 + t)} = 4 \int \frac{t + 1 - 1}{(1 + t)} d t

\Rightarrow Ω = 4 \int d t - 4 \int \frac{1}{(1 + t)} d t

\Rightarrow Ω = 4 t - 4 l n ∣ 1 + t ∣ + C

\Rightarrow Ω = 4 \sqrt{u} - 4 l n ∣ 1 + \sqrt{u} ∣ + C

∴∵ Ω = 4 (\sqrt[4]{x} - l n ∣ 1 + \sqrt[4]{x} ∣ + C

\dots \dots \dots \dots L e p u i s s a n t \dots \dots \dots \dots

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