What is the integral of x^5sqrt(15+x^6) ?

The integral of x^5sqrt(15+x^6) is 1/9 (15+x^6)^{3/2}+C

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integral of x^5sqrt(15+x^6)

\int x^{5} 15 + x^{6} d x

\frac{1}{9} (15 + x^{6})^{\frac{3}{2}} + C

Solution steps

\int x^{5} 15 + x^{6} d x

Apply u-substitution

: \int \frac{u}{6} d u

= \int \frac{u}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = 15 + x^{6}

= \frac{1}{6} \cdot \frac{2}{3} (15 + x^{6})^{\frac{3}{2}}

Simplify

\frac{1}{6} \cdot \frac{2}{3} (15 + x^{6})^{\frac{3}{2}} : \frac{1}{9} (15 + x^{6})^{\frac{3}{2}}

= \frac{1}{9} (15 + x^{6})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{9} (15 + x^{6})^{\frac{3}{2}} + C

\frac{\partial}{\partial z} (x e^{- z})

\frac{\partial}{\partial z} (x sin (y - z))

x \to 0 lim (5 e)

\frac{d}{d x} (\frac{9}{x + 7})

\frac{\partial}{\partial x} (sin (α) cos (β))

What is the integral of x^5sqrt(15+x^6) ?
The integral of x^5sqrt(15+x^6) is 1/9 (15+x^6)^{3/2}+C