What is the integral of (x^2-6)^6*x ?

The integral of (x^2-6)^6*x is 1/14 (x^2-6)^7+C

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integral of (x^2-6)^6*x

\int (x^{2} - 6)^{6} \cdot x d x

\frac{1}{14} (x^{2} - 6)^{7} + C

Solution steps

\int (x^{2} - 6)^{6} x d x

Apply u-substitution

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

Apply the Power Rule

= \frac{1}{2} \cdot \frac{u ^{7}}{7}

Substitute back

u = x^{2} - 6

= \frac{1}{2} \cdot \frac{( x ^{2} - 6 ) ^{7}}{7}

Simplify

\frac{1}{2} \cdot \frac{( x ^{2} - 6 ) ^{7}}{7} : \frac{1}{14} (x^{2} - 6)^{7}

= \frac{1}{14} (x^{2} - 6)^{7}

Add a constant to the solution

= \frac{1}{14} (x^{2} - 6)^{7} + C

What is the integral of (x^2-6)^6*x ?
The integral of (x^2-6)^6*x is 1/14 (x^2-6)^7+C