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

The integral of (2x+7)(x^2+7x+1) is (2x^4)/3-(x^4)/6+7x^3+x^2+(49x^2)/2+7x+C

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integral of (2x+7)(x^2+7x+1)

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

\frac{2 x ^{4}}{3} - \frac{x ^{4}}{6} + 7 x^{3} + x^{2} + \frac{49 x ^{2}}{2} + 7 x + C

Solution steps

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

Apply Integration By Parts

: \frac{2 x ^{4}}{3} + 7 x^{3} + 2 x^{2} + \frac{7 x ^{3}}{3} + \frac{49 x ^{2}}{2} + 7 x - \int \frac{2 x ^{3}}{3} + 7 x^{2} + 2 x d x

= \frac{2 x ^{4}}{3} + 7 x^{3} + 2 x^{2} + \frac{7 x ^{3}}{3} + \frac{49 x ^{2}}{2} + 7 x - \int \frac{2 x ^{3}}{3} + 7 x^{2} + 2 x d x

\int \frac{2 x ^{3}}{3} + 7 x^{2} + 2 x d x = \frac{x ^{4}}{6} + \frac{7 x ^{3}}{3} + x^{2}

= \frac{2 x ^{4}}{3} + 7 x^{3} + 2 x^{2} + \frac{7 x ^{3}}{3} + \frac{49 x ^{2}}{2} + 7 x - (\frac{x ^{4}}{6} + \frac{7 x ^{3}}{3} + x^{2})

Simplify

\frac{2 x ^{4}}{3} + 7 x^{3} + 2 x^{2} + \frac{7 x ^{3}}{3} + \frac{49 x ^{2}}{2} + 7 x - (\frac{x ^{4}}{6} + \frac{7 x ^{3}}{3} + x^{2}) : \frac{2 x ^{4}}{3} - \frac{x ^{4}}{6} + 7 x^{3} + x^{2} + \frac{49 x ^{2}}{2} + 7 x

= \frac{2 x ^{4}}{3} - \frac{x ^{4}}{6} + 7 x^{3} + x^{2} + \frac{49 x ^{2}}{2} + 7 x

Add a constant to the solution

= \frac{2 x ^{4}}{3} - \frac{x ^{4}}{6} + 7 x^{3} + x^{2} + \frac{49 x ^{2}}{2} + 7 x + C

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What is the integral of (2x+7)(x^2+7x+1) ?
The integral of (2x+7)(x^2+7x+1) is (2x^4)/3-(x^4)/6+7x^3+x^2+(49x^2)/2+7x+C