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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int 2 v^{2} 1 + v^{2} d v

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

Take the constant out:

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

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

Apply Trigonometric Substitution

: \int tan^{2} (w) sec^{2} (w) sec^{2} (w) d w

= \frac{3}{2} \cdot 2 \cdot \int tan^{2} (w) sec^{2} (w) sec^{2} (w) d w

sec^{2} (w) = (sec (w)),

assuming

sec (w) \geq 0

= \frac{3}{2} \cdot 2 \cdot \int tan^{2} (w) sec^{2} (w) sec (w) d w

Simplify

tan^{2} (w) sec^{2} (w) sec (w) : sec^{3} (w) tan^{2} (w)

= \frac{3}{2} \cdot 2 \cdot \int sec^{3} (w) tan^{2} (w) d w

Rewrite using trig identities

= \frac{3}{2} \cdot 2 \cdot \int sec^{3} (w) (- 1 + sec^{2} (w)) d w

Expand

sec^{3} (w) (- 1 + sec^{2} (w)) : - sec^{3} (w) + sec^{5} (w)

= \frac{3}{2} \cdot 2 \cdot \int - sec^{3} (w) + sec^{5} (w) d w

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{3}{2} \cdot 2 (- \int sec^{3} (w) d w + \int sec^{5} (w) d w)

\int sec^{3} (w) d w = \frac{1}{2} sec (w) tan (w) + \frac{1}{2} ln ∣ tan (w) + sec (w) ∣

\int sec^{5} (w) d w = \frac{sec ^{4} ( w ) sin ( w )}{4} + \frac{3}{4} (\frac{1}{2} sec (w) tan (w) + \frac{1}{2} ln ∣ tan (w) + sec (w) ∣)

= \frac{3}{2} \cdot 2 (- (\frac{1}{2} sec (w) tan (w) + \frac{1}{2} ln ∣ tan (w) + sec (w) ∣) + \frac{sec ^{4} ( w ) sin ( w )}{4} + \frac{3}{4} (\frac{1}{2} sec (w) tan (w) + \frac{1}{2} ln ∣ tan (w) + sec (w) ∣))

Substitute back

= \frac{3}{2} \cdot 2 - (\frac{1}{2} sec (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})) tan (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})) + \frac{1}{2} ln tan (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})) + sec (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}}))) + \frac{sec ^{4} ( arctan ( ( x ^{\frac{2}{3}} ) ^{\frac{1}{2}} ) ) sin ( arctan ( ( x ^{\frac{2}{3}} ) ^{\frac{1}{2}} ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})) tan (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})) + \frac{1}{2} ln tan (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})) + sec (arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})))

Simplify

Add a constant to the solution

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