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

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

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

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

\frac{1}{9} (5 + x^{6})^{\frac{3}{2}} + C

Solution steps

\int x^{5} 5 + 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 = 5 + x^{6}

= \frac{1}{6} \cdot \frac{2}{3} (5 + x^{6})^{\frac{3}{2}}

Simplify

\frac{1}{6} \cdot \frac{2}{3} (5 + x^{6})^{\frac{3}{2}} : \frac{1}{9} (5 + x^{6})^{\frac{3}{2}}

= \frac{1}{9} (5 + x^{6})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{9} (5 + x^{6})^{\frac{3}{2}} + C

\int \frac{x ^{3} - 6 x + 8}{x ^{2}} d x

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

(e^{4 x})^{^{'}}

\frac{d}{d x} (5 \cdot e^{x} + 2 \cdot 3^{x})

e^{- t^{2}}

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