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

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

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

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

2 (- \frac{1}{3} ln x^{\frac{1}{2}} + 1 + \frac{1}{6} (ln 4 x - 4 x^{\frac{1}{2}} + 4 + 23 arctan (\frac{1}{3} (2 x^{\frac{1}{2}} - 1)))) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{3 v}{1 + v ^{3}} d v

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

Take the constant out:

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

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

Take the partial fraction of

\frac{v}{1 + v ^{3}} : - \frac{1}{3 ( v + 1 )} + \frac{v + 1}{3 ( v ^{2} - v + 1 )}

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

Apply the Sum Rule:

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

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

\int \frac{1}{3 ( v + 1 )} d v = \frac{1}{3} ln ∣ v + 1 ∣

\int \frac{v + 1}{3 ( v ^{2} - v + 1 )} d v = \frac{1}{6} (ln 4 v^{2} - 4 v + 4 + 23 arctan (\frac{2 v - 1}{3}))

= \frac{2}{3} \cdot 3 (- \frac{1}{3} ln ∣ v + 1 ∣ + \frac{1}{6} (ln 4 v^{2} - 4 v + 4 + 23 arctan (\frac{2 v - 1}{3})))

Substitute back

= \frac{2}{3} \cdot 3 - \frac{1}{3} ln (x^{\frac{3}{2}})^{\frac{1}{3}} + 1 + \frac{1}{6} ln 4 ((x^{\frac{3}{2}})^{\frac{1}{3}})^{2} - 4 (x^{\frac{3}{2}})^{\frac{1}{3}} + 4 + 23 arctan \frac{2 ( x ^{\frac{3}{2}} ) ^{\frac{1}{3}} - 1}{3}

Simplify

\frac{2}{3} \cdot 3 - \frac{1}{3} ln (x^{\frac{3}{2}})^{\frac{1}{3}} + 1 + \frac{1}{6} ln 4 ((x^{\frac{3}{2}})^{\frac{1}{3}})^{2} - 4 (x^{\frac{3}{2}})^{\frac{1}{3}} + 4 + 23 arctan \frac{2 ( x ^{\frac{3}{2}} ) ^{\frac{1}{3}} - 1}{3} : 2 (- \frac{1}{3} ln x^{\frac{1}{2}} + 1 + \frac{1}{6} (ln 4 x - 4 x^{\frac{1}{2}} + 4 + 23 arctan (\frac{1}{3} (2 x^{\frac{1}{2}} - 1))))

= 2 (- \frac{1}{3} ln x^{\frac{1}{2}} + 1 + \frac{1}{6} (ln 4 x - 4 x^{\frac{1}{2}} + 4 + 23 arctan (\frac{1}{3} (2 x^{\frac{1}{2}} - 1))))

Add a constant to the solution

= 2 (- \frac{1}{3} ln x^{\frac{1}{2}} + 1 + \frac{1}{6} (ln 4 x - 4 x^{\frac{1}{2}} + 4 + 23 arctan (\frac{1}{3} (2 x^{\frac{1}{2}} - 1)))) + C

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