`int sqrt(x^3) root(3)(x^2)` Find the general indefinite integral.

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Chapter 5, 5.4 - Problem 6 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`int sqrt(x^3)root(3)(x^2) dx`

Before evaluating, convert the radicals to expressions with rational exponents.

`= int x^(3/2)*x^(2/3) dx`

Then, simplify the integrand. Apply the laws of exponent `a^m *a^n =a^(m+n)` .

`= int x^(3/2+2/3) dx`

`= int x^(9/6+4/6)dx`

`= int x^(13/6)dx`

Then, apply the integral formula `int x^n dx = x^(n+1)/(n+1) + C` .

`= x^(13/6+1)/(13/6+1) + C`

`= x^(19/6)/(19/6) + C`

`= (6x^(19/6))/19 + C`

 

Therefore,  `int sqrt(x^3)root(3)(x^2)dx = (6x^(19/6))/19 + C` .

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