What is the integral of sqrt(1+x^{-2/3)} ?

The integral of sqrt(1+x^{-2/3)} is (x^{2/3}+1)^{3/2}+C

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integral of sqrt(1+x^{-2/3)}

\int 1 + x^{- \frac{2}{3}} d x

(x^{\frac{2}{3}} + 1)^{\frac{3}{2}} + C

Solution steps

\int 1 + x^{- \frac{2}{3}} d x

Simplify

1 + x^{- \frac{2}{3}} : \frac{x ^{\frac{2}{3}} + 1}{x ^{\frac{2}{3}}}

= \int \frac{x ^{\frac{2}{3}} + 1}{x ^{\frac{2}{3}}} d x

Apply u-substitution

: \int \frac{3 u ^{\frac{1}{2}}}{2} d u

= \int \frac{3 u ^{\frac{1}{2}}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{3}{2} \cdot \int u^{\frac{1}{2}} d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{3}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x^{\frac{2}{3}} + 1

= \frac{3}{2} \cdot \frac{2}{3} (x^{\frac{2}{3}} + 1)^{\frac{3}{2}}

Simplify

\frac{3}{2} \cdot \frac{2}{3} (x^{\frac{2}{3}} + 1)^{\frac{3}{2}} : (x^{\frac{2}{3}} + 1)^{\frac{3}{2}}

= (x^{\frac{2}{3}} + 1)^{\frac{3}{2}}

Add a constant to the solution

= (x^{\frac{2}{3}} + 1)^{\frac{3}{2}} + C

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What is the integral of sqrt(1+x^{-2/3)} ?
The integral of sqrt(1+x^{-2/3)} is (x^{2/3}+1)^{3/2}+C