Solveeit Logo
Login

Question

Question: The value of integral \(\int_{}^{}\frac{dx}{x\sqrt{1 - x^{3}}}\) is given by –...

The value of integral ∫dxx1−x3\int_{}^{}\frac{dx}{x\sqrt{1 - x^{3}}} is given by –

A

13\frac{1}{3}log∣1−x3+11−x3−1∣\left| \frac{\sqrt{1 - x^{3}} + 1}{\sqrt{1 - x^{3}} - 1} \right| + c

B

13log⁡∣1−x3–11−x3+1∣\frac{1}{3}\log\left| \frac{\sqrt{1 - x^{3}}–1}{\sqrt{1 - x^{3}} + 1} \right|+ c

C

23log⁡∣11−x3∣\frac{2}{3}\log\left| \frac{1}{\sqrt{1 - x^{3}}} \right| + c

D

13\frac{1}{3} log |1– x3| + c

Answer

13log⁡∣1−x3–11−x3+1∣\frac{1}{3}\log\left| \frac{\sqrt{1 - x^{3}}–1}{\sqrt{1 - x^{3}} + 1} \right|+ c

Explanation

Solution

Put 1 – x3 = t2, Then –3x2dx = 2t dt and the integral becomes

– 13\frac{1}{3} ∫−3x2dxx31−x3\int_{}^{}\frac{- 3x^{2}dx}{x^{3}\sqrt{1 - x^{3}}}= – 13\frac{1}{3} ∫2tdt(1−t2)t\int_{}^{}\frac{2tdt}{(1 - t^{2})t} = 23\frac{2}{3} ∫dtt2−1\int_{}^{}\frac{dt}{t^{2} - 1}

= 23\frac{2}{3} (12log⁡∣t−1t+1∣)\left( \frac{1}{2}\log\left| \frac{t - 1}{t + 1} \right| \right) + c = 13log⁡∣1−x3–11−x3+1∣\frac{1}{3}\log\left| \frac{\sqrt{1 - x^{3}}–1}{\sqrt{1 - x^{3}} + 1} \right|+ c.