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Question

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

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

A

3x4/3+73x7/3+c3x^{4/3} + \frac{7}{3}x^{7/3} + c

B

x4/3+53x5/3+cx^{4/3} + \frac{5}{3}x^{5/3} + c

C

3x4/3+35x5/3+c3x^{4/3} + \frac{3}{5}x^{5/3} + c

D

3x4/3+37x7/3+2411x11/6+c3x^{4/3} + \frac{3}{7}x^{7/3} + \frac{24}{11}x^{11/6} + c

Answer

3x4/3+37x7/3+2411x11/6+c3x^{4/3} + \frac{3}{7}x^{7/3} + \frac{24}{11}x^{11/6} + c

Explanation

Solution

I=∫x1/3(2+x1/2)2dxI = \int_{}^{}{x^{1/3}(2 + x^{1/2})^{2}dx}

Since P is natural number,

I=∫x1/3(4+x+4x1/2)dx=∫(4x1/3+x4/3+4x5/6)dxI = \int_{}^{}{x^{1/3}(4 + x + 4x^{1/2})dx} = \int_{}^{}{(4x^{1/3} + x^{4/3} + 4x^{5/6})dx} =4x4/34/3+x7/37/3+4.x11/611/6+c= \frac{4x^{4/3}}{4/3} + \frac{x^{7/3}}{7/3} + \frac{4.x^{11/6}}{11/6} + c =3x4/3+37x7/3+2411x11/6+c.= 3x^{4/3} + \frac{3}{7}x^{7/3} + \frac{24}{11}x^{11/6} + c.