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

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integral of 3/(2+x^3)

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

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

Solution steps

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

Take the constant out:

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

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

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

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

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

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

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

= 3 \cdot \frac{1}{2} 32 \cdot \int \frac{1}{1 + u ^{3}} d u

Take the partial fraction of

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

= 3 \cdot \frac{1}{2} 32 \cdot \int \frac{1}{3 ( u + 1 )} + \frac{- u + 2}{3 ( u ^{2} - u + 1 )} d u

Apply the Sum Rule:

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

= 3 \cdot \frac{1}{2} 32 (\int \frac{1}{3 ( u + 1 )} d u + \int \frac{- u + 2}{3 ( u ^{2} - u + 1 )} d u)

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

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

= 3 \cdot \frac{1}{2} 32 (\frac{1}{3} ln ∣ u + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 u ^{2} - 4 u}{3} + 23 arctan (\frac{2 u - 1}{3})))

Substitute back

u = \frac{x}{3 2}

= 3 \cdot \frac{1}{2} 32 \frac{1}{3} ln \frac{x}{3 2} + 1 + \frac{1}{6} - ln \frac{4}{3} + \frac{4 ( \frac{x}{3 2} ) ^{2} - 4 \cdot \frac{x}{3 2}}{3} + 23 arctan (\frac{2 \cdot \frac{x}{3 2} - 1}{3})

Simplify

3 \cdot \frac{1}{2} 32 \frac{1}{3} ln \frac{x}{3 2} + 1 + \frac{1}{6} - ln \frac{4}{3} + \frac{4 ( \frac{x}{3 2} ) ^{2} - 4 \cdot \frac{x}{3 2}}{3} + 23 arctan (\frac{2 \cdot \frac{x}{3 2} - 1}{3}) : \frac{1}{2 \cdot 2 ^{\frac{2}{3}}} (2 ln \frac{x}{3 2} + 1 + 23 arctan (\frac{2 ^{\frac{2}{3}} x - 1}{3}) - ln \frac{2 3 2 x ^{2} - 2 ^{1 + \frac{2}{3}} x}{3} + \frac{4}{3})

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

Add a constant to the solution

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

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