What is the integral of x^2sqrt(x^3+7) ?

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

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integral of x^2sqrt(x^3+7)

\int x^{2} x^{3} + 7 d x

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

Solution steps

\int x^{2} x^{3} + 7 d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{3} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = x^{3} + 7

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

Simplify

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

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

Add a constant to the solution

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

\frac{d y}{d x}, 2 x^{2} + 2 x + x y = 1

f (x) = e^{x} - x^{e}

\frac{\partial}{\partial x} (A^{2} x^{2} + B^{2} y^{2})

\frac{d}{d x} (\frac{lo g _{10} ( x )}{x})

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