What is the integral of 2(2x+6)^7 ?

The integral of 2(2x+6)^7 is 1/8 (2x+6)^8+C

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integral of 2(2x+6)^7

\int 2 (2 x + 6)^{7} d x

\frac{1}{8} (2 x + 6)^{8} + C

Solution steps

\int 2 (2 x + 6)^{7} d x

Take the constant out:

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

= 2 \cdot \int (2 x + 6)^{7} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{8}}{8}

Substitute back

u = 2 x + 6

= 2 \cdot \frac{1}{2} \cdot \frac{( 2 x + 6 ) ^{8}}{8}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{( 2 x + 6 ) ^{8}}{8} : \frac{1}{8} (2 x + 6)^{8}

= \frac{1}{8} (2 x + 6)^{8}

Add a constant to the solution

= \frac{1}{8} (2 x + 6)^{8} + C