What is the integral of x^9sqrt(x^5+6) ?

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

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integral of x^9sqrt(x^5+6)

\int x^{9} x^{5} + 6 d x

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

Solution steps

\int x^{9} x^{5} + 6 d x

Apply u-substitution

: \int \frac{2 ( u ^{2} - 6 ) u ^{2}}{5} d u

= \int \frac{2 ( u ^{2} - 6 ) u ^{2}}{5} d u

Take the constant out:

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

= \frac{2}{5} \cdot \int (u^{2} - 6) u^{2} d u

Expand

(u^{2} - 6) u^{2} : u^{4} - 6 u^{2}

= \frac{2}{5} \cdot \int u^{4} - 6 u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{2}{5} (\int u^{4} d u - \int 6 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 6 u^{2} d u = 2 u^{3}

= \frac{2}{5} (\frac{u ^{5}}{5} - 2 u^{3})

Substitute back

u = x^{5} + 6

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

Simplify

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

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

Add a constant to the solution

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

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\int \frac{x ^{4}}{( 2 - x ^{5} ) ^{4}} d x

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