What is the integral of (x^2+4x+10)(2x+4) ?

The integral of (x^2+4x+10)(2x+4) is (x^4)/2+4x^3+18x^2+40x+C

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integral of (x^2+4x+10)(2x+4)

\int (x^{2} + 4 x + 10) (2 x + 4) d x

\frac{x ^{4}}{2} + 4 x^{3} + 18 x^{2} + 40 x + C

Solution steps

\int (x^{2} + 4 x + 10) (2 x + 4) d x

Expand

(x^{2} + 4 x + 10) (2 x + 4) : 2 x^{3} + 12 x^{2} + 36 x + 40

= \int 2 x^{3} + 12 x^{2} + 36 x + 40 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

= \int 2 x^{3} d x + \int 12 x^{2} d x + \int 36 x d x + \int 40 d x

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

\int 12 x^{2} d x = 4 x^{3}

\int 36 x d x = 18 x^{2}

\int 40 d x = 40 x

= \frac{x ^{4}}{2} + 4 x^{3} + 18 x^{2} + 40 x

Add a constant to the solution

= \frac{x ^{4}}{2} + 4 x^{3} + 18 x^{2} + 40 x + C

\int 4 x (x^{2} + 1) d x

\int 4 x (x^{2} + 2)^{3} d x

\int \frac{x}{y} d (\frac{x}{y})

\int \frac{1}{z ^{2} - 2 z} d z

\int y (4 - y^{2})^{\frac{3}{2}} d y

What is the integral of (x^2+4x+10)(2x+4) ?
The integral of (x^2+4x+10)(2x+4) is (x^4)/2+4x^3+18x^2+40x+C