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

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

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integral of (3sqrt(x))/(1+x^3)

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

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

Solution steps

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

Apply u-substitution

= 3 \cdot \int \frac{2 u ^{2}}{1 + u ^{6}} d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int \frac{u ^{2}}{1 + u ^{6}} d u

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= 3 \cdot 2 \cdot \frac{1}{3} arctan (v)

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

y^{^{'}} + 2 x y = y + 4 x - 2

\frac{d}{d x} (\frac{1}{1 + e ^{x}})

\frac{d}{d x} (\frac{sin ( x + 1 )}{e ^{x}})

in v erse l a pl a ce \frac{e ^{- 4 x}}{( x - 2 ) ^{2}}

x \to \infty lim ((1 + \frac{c}{x})^{2 x})

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