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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x^{3}

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

Simplify

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

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

Add a constant to the solution

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

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