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Question

Question: Evaluate \(\int_{}^{}{x^{- 2/3}(1 + x^{1/3})^{1/2}dx}\)...

Evaluate ∫x−2/3(1+x1/3)1/2dx\int_{}^{}{x^{- 2/3}(1 + x^{1/3})^{1/2}dx}

A

2(1+x1/3)2/3+c2(1 + x^{1/3})^{2/3} + c

B

2(1+x1/3)3/2+c2(1 + x^{1/3})^{3/2} + c

C

2(1+x2/3)3/2+c2(1 + x^{2/3})^{3/2} + c

D

2(1+x2/3)2/3+c2(1 + x^{2/3})^{2/3} + c

Answer

2(1+x1/3)3/2+c2(1 + x^{1/3})^{3/2} + c

Explanation

Solution

If we substitute 1=x1/3=t21 = x^{1/3} = t^{2}then, 13x2/3dx=dt\frac{1}{3x^{2/3}}dx = dt

∴I=∫t.6tdt1=6∫t2dt\therefore I = \int_{}^{}{\frac{t.6tdt}{1} = 6\int_{}^{}{t^{2}dt}} =2t3+c= 2t^{3} + c or I=2(1+x1/3)3/2+c.I = 2(1 + x^{1/3})^{3/2} + c.