What is the integral of 2x(3x-1)^7 ?

The integral of 2x(3x-1)^7 is 486x^9-(5103x^8)/4+1458x^7-945x^6+378x^5-(189x^4)/2+14x^3-x^2+C

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integral of 2x(3x-1)^7

\int 2 x (3 x - 1)^{7} d x

486 x^{9} - \frac{5103 x ^{8}}{4} + 1458 x^{7} - 945 x^{6} + 378 x^{5} - \frac{189 x ^{4}}{2} + 14 x^{3} - x^{2} + C

Solution steps

\int 2 x (3 x - 1)^{7} d x

Take the constant out:

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

= 2 \cdot \int x (3 x - 1)^{7} d x

Expand

x (3 x - 1)^{7} : 2187 x^{8} - 5103 x^{7} + 5103 x^{6} - 2835 x^{5} + 945 x^{4} - 189 x^{3} + 21 x^{2} - x

= 2 \cdot \int 2187 x^{8} - 5103 x^{7} + 5103 x^{6} - 2835 x^{5} + 945 x^{4} - 189 x^{3} + 21 x^{2} - 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

= 2 (\int 2187 x^{8} d x - \int 5103 x^{7} d x + \int 5103 x^{6} d x - \int 2835 x^{5} d x + \int 945 x^{4} d x - \int 189 x^{3} d x + \int 21 x^{2} d x - \int x d x)

\int 2187 x^{8} d x = 243 x^{9}

\int 5103 x^{7} d x = \frac{5103 x ^{8}}{8}

\int 5103 x^{6} d x = 729 x^{7}

\int 2835 x^{5} d x = \frac{945 x ^{6}}{2}

\int 945 x^{4} d x = 189 x^{5}

\int 189 x^{3} d x = \frac{189 x ^{4}}{4}

\int 21 x^{2} d x = 7 x^{3}

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

= 2 (243 x^{9} - \frac{5103 x ^{8}}{8} + 729 x^{7} - \frac{945 x ^{6}}{2} + 189 x^{5} - \frac{189 x ^{4}}{4} + 7 x^{3} - \frac{x ^{2}}{2})

Simplify

2 (243 x^{9} - \frac{5103 x ^{8}}{8} + 729 x^{7} - \frac{945 x ^{6}}{2} + 189 x^{5} - \frac{189 x ^{4}}{4} + 7 x^{3} - \frac{x ^{2}}{2}) : 486 x^{9} - \frac{5103 x ^{8}}{4} + 1458 x^{7} - 945 x^{6} + 378 x^{5} - \frac{189 x ^{4}}{2} + 14 x^{3} - x^{2}

= 486 x^{9} - \frac{5103 x ^{8}}{4} + 1458 x^{7} - 945 x^{6} + 378 x^{5} - \frac{189 x ^{4}}{2} + 14 x^{3} - x^{2}

Add a constant to the solution

= 486 x^{9} - \frac{5103 x ^{8}}{4} + 1458 x^{7} - 945 x^{6} + 378 x^{5} - \frac{189 x ^{4}}{2} + 14 x^{3} - x^{2} + C

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What is the integral of 2x(3x-1)^7 ?
The integral of 2x(3x-1)^7 is 486x^9-(5103x^8)/4+1458x^7-945x^6+378x^5-(189x^4)/2+14x^3-x^2+C