What is the integral of x(x^2+6)^9 ?

The integral of x(x^2+6)^9 is 1/20 (x^2+6)^{10}+C

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integral of x(x^2+6)^9

\int x (x^{2} + 6)^{9} d x

\frac{1}{20} (x^{2} + 6)^{10} + C

Solution steps

\int x (x^{2} + 6)^{9} d x

Apply u-substitution

= \int \frac{u ^{9}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u^{9} d u

Apply the Power Rule

= \frac{1}{2} \cdot \frac{u ^{10}}{10}

Substitute back

u = x^{2} + 6

= \frac{1}{2} \cdot \frac{( x ^{2} + 6 ) ^{10}}{10}

Simplify

\frac{1}{2} \cdot \frac{( x ^{2} + 6 ) ^{10}}{10} : \frac{1}{20} (x^{2} + 6)^{10}

= \frac{1}{20} (x^{2} + 6)^{10}

Add a constant to the solution

= \frac{1}{20} (x^{2} + 6)^{10} + C

What is the integral of x(x^2+6)^9 ?
The integral of x(x^2+6)^9 is 1/20 (x^2+6)^{10}+C