What is the integral of 12(x-4)^5 ?

The integral of 12(x-4)^5 is 2(x-4)^6+C

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integral of 12(x-4)^5

\int 12 (x - 4)^{5} d x

2 (x - 4)^{6} + C

Solution steps

\int 12 (x - 4)^{5} d x

Take the constant out:

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

= 12 \cdot \int (x - 4)^{5} d x

Apply u-substitution

= 12 \cdot \int u^{5} d u

Apply the Power Rule

= 12 \cdot \frac{u ^{6}}{6}

Substitute back

u = x - 4

= 12 \cdot \frac{( x - 4 ) ^{6}}{6}

Simplify

12 \cdot \frac{( x - 4 ) ^{6}}{6} : 2 (x - 4)^{6}

= 2 (x - 4)^{6}

Add a constant to the solution

= 2 (x - 4)^{6} + C