What is the integral of sqrt(x)(sqrt(1+x\sqrt{x)}) ?

The integral of sqrt(x)(sqrt(1+x\sqrt{x)}) is 4/9 (1+xsqrt(x))^{3/2}+C

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integral of sqrt(x)(sqrt(1+x\sqrt{x)})

\int x (1 + x x) d x

\frac{4}{9} (1 + x x)^{\frac{3}{2}} + C

Solution steps

\int x 1 + x x d x

Apply u-substitution

: \int \frac{2 u}{3} d u

= \int \frac{2 u}{3} 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 u d u

Apply the Power Rule

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

= \frac{2}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + x x

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

Simplify

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

= \frac{4}{9} (1 + x x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{9} (1 + x x)^{\frac{3}{2}} + C

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What is the integral of sqrt(x)(sqrt(1+x\sqrt{x)}) ?
The integral of sqrt(x)(sqrt(1+x\sqrt{x)}) is 4/9 (1+xsqrt(x))^{3/2}+C