What is the integral of x^9sqrt(x^5+7) ?

The integral of x^9sqrt(x^5+7) is 2/5 (1/5 (x^5+7)^{5/2}-7/3 (x^5+7)^{3/2})+C

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integral of x^9sqrt(x^5+7)

\int x^{9} x^{5} + 7 d x

\frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}}) + C

Solution steps

\int x^{9} x^{5} + 7 d x

Apply u-substitution

= \int \frac{2 ( u ^{2} - 7 ) 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} - 7) u^{2} d u

Expand

(u^{2} - 7) u^{2} : u^{4} - 7 u^{2}

= \frac{2}{5} \cdot \int u^{4} - 7 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 7 u^{2} d u)

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

\int 7 u^{2} d u = \frac{7 u ^{3}}{3}

= \frac{2}{5} (\frac{u ^{5}}{5} - \frac{7 u ^{3}}{3})

Substitute back

u = x^{5} + 7

= \frac{2}{5} (\frac{( x ^{5} + 7 ) ^{5}}{5} - \frac{7 ( x ^{5} + 7 ) ^{3}}{3})

Simplify

\frac{2}{5} (\frac{( x ^{5} + 7 ) ^{5}}{5} - \frac{7 ( x ^{5} + 7 ) ^{3}}{3}) : \frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}})

= \frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}})

Add a constant to the solution

= \frac{2}{5} (\frac{1}{5} (x^{5} + 7)^{\frac{5}{2}} - \frac{7}{3} (x^{5} + 7)^{\frac{3}{2}}) + C

What is the integral of x^9sqrt(x^5+7) ?
The integral of x^9sqrt(x^5+7) is 2/5 (1/5 (x^5+7)^{5/2}-7/3 (x^5+7)^{3/2})+C