What is the integral of sqrt(x)sqrt(x\sqrt{x)+7} ?

The integral of sqrt(x)sqrt(x\sqrt{x)+7} is 4/9 (x^{3/2}+7)^{3/2}+C

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integral of sqrt(x)sqrt(x\sqrt{x)+7}

\int x x x + 7 d x

\frac{4}{9} (x^{\frac{3}{2}} + 7)^{\frac{3}{2}} + C

Solution steps

\int x x x + 7 d x

Apply u-substitution

: \int 2 u^{2} u^{3} + 7 d u

= \int 2 u^{2} u^{3} + 7 d u

Take the constant out:

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

= 2 \cdot \int u^{2} u^{3} + 7 d u

Apply u-substitution

: \int \frac{v}{3} d v

= 2 \cdot \int \frac{v}{3} d v

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

= 2 \cdot \frac{1}{3} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

= 2 \cdot \frac{1}{3} \cdot \frac{2}{3} ((x)^{3} + 7)^{\frac{3}{2}}

Simplify

2 \cdot \frac{1}{3} \cdot \frac{2}{3} ((x)^{3} + 7)^{\frac{3}{2}} : \frac{4}{9} (x^{\frac{3}{2}} + 7)^{\frac{3}{2}}

= \frac{4}{9} (x^{\frac{3}{2}} + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{9} (x^{\frac{3}{2}} + 7)^{\frac{3}{2}} + C

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What is the integral of sqrt(x)sqrt(x\sqrt{x)+7} ?
The integral of sqrt(x)sqrt(x\sqrt{x)+7} is 4/9 (x^{3/2}+7)^{3/2}+C