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

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

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integral of (x^2+2x+1)(3x+1)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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