What is the integral of 4x(x+2)^7 ?

The integral of 4x(x+2)^7 is (4x^9)/9+7x^8+48x^7+(560x^6)/3+448x^5+672x^4+(1792x^3)/3+256x^2+C

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integral of 4x(x+2)^7

\int 4 x (x + 2)^{7} d x

\frac{4 x ^{9}}{9} + 7 x^{8} + 48 x^{7} + \frac{560 x ^{6}}{3} + 448 x^{5} + 672 x^{4} + \frac{1792 x ^{3}}{3} + 256 x^{2} + C

Solution steps

\int 4 x (x + 2)^{7} d x

Take the constant out:

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

= 4 \cdot \int x (x + 2)^{7} d x

Expand

x (x + 2)^{7} : x^{8} + 14 x^{7} + 84 x^{6} + 280 x^{5} + 560 x^{4} + 672 x^{3} + 448 x^{2} + 128 x

= 4 \cdot \int x^{8} + 14 x^{7} + 84 x^{6} + 280 x^{5} + 560 x^{4} + 672 x^{3} + 448 x^{2} + 128 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int x^{8} d x + \int 14 x^{7} d x + \int 84 x^{6} d x + \int 280 x^{5} d x + \int 560 x^{4} d x + \int 672 x^{3} d x + \int 448 x^{2} d x + \int 128 x d x)

\int x^{8} d x = \frac{x ^{9}}{9}

\int 14 x^{7} d x = \frac{7 x ^{8}}{4}

\int 84 x^{6} d x = 12 x^{7}

\int 280 x^{5} d x = \frac{140 x ^{6}}{3}

\int 560 x^{4} d x = 112 x^{5}

\int 672 x^{3} d x = 168 x^{4}

\int 448 x^{2} d x = \frac{448 x ^{3}}{3}

\int 128 x d x = 64 x^{2}

= 4 (\frac{x ^{9}}{9} + \frac{7 x ^{8}}{4} + 12 x^{7} + \frac{140 x ^{6}}{3} + 112 x^{5} + 168 x^{4} + \frac{448 x ^{3}}{3} + 64 x^{2})

Simplify

4 (\frac{x ^{9}}{9} + \frac{7 x ^{8}}{4} + 12 x^{7} + \frac{140 x ^{6}}{3} + 112 x^{5} + 168 x^{4} + \frac{448 x ^{3}}{3} + 64 x^{2}) : \frac{4 x ^{9}}{9} + 7 x^{8} + 48 x^{7} + \frac{560 x ^{6}}{3} + 448 x^{5} + 672 x^{4} + \frac{1792 x ^{3}}{3} + 256 x^{2}

= \frac{4 x ^{9}}{9} + 7 x^{8} + 48 x^{7} + \frac{560 x ^{6}}{3} + 448 x^{5} + 672 x^{4} + \frac{1792 x ^{3}}{3} + 256 x^{2}

Add a constant to the solution

= \frac{4 x ^{9}}{9} + 7 x^{8} + 48 x^{7} + \frac{560 x ^{6}}{3} + 448 x^{5} + 672 x^{4} + \frac{1792 x ^{3}}{3} + 256 x^{2} + C

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What is the integral of 4x(x+2)^7 ?
The integral of 4x(x+2)^7 is (4x^9)/9+7x^8+48x^7+(560x^6)/3+448x^5+672x^4+(1792x^3)/3+256x^2+C