What is the integral of 10x^3+2x^2+1/2 x+3 ?

The integral of 10x^3+2x^2+1/2 x+3 is (5x^4)/2+(2x^3)/3+(x^2)/4+3x+C

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integral of 10x^3+2x^2+1/2 x+3

\int 10 x^{3} + 2 x^{2} + \frac{1}{2} x + 3 d x

\frac{5 x ^{4}}{2} + \frac{2 x ^{3}}{3} + \frac{x ^{2}}{4} + 3 x + C

Solution steps

\int 10 x^{3} + 2 x^{2} + \frac{1}{2} x + 3 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 10 x^{3} d x + \int 2 x^{2} d x + \int \frac{1}{2} x d x + \int 3 d x

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

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

\int \frac{1}{2} x d x = \frac{x ^{2}}{4}

\int 3 d x = 3 x

= \frac{5 x ^{4}}{2} + \frac{2 x ^{3}}{3} + \frac{x ^{2}}{4} + 3 x

Add a constant to the solution

= \frac{5 x ^{4}}{2} + \frac{2 x ^{3}}{3} + \frac{x ^{2}}{4} + 3 x + C

\frac{d y}{d x} = (y - 5)^{2}

\frac{d y}{d x}

\frac{- 14 x}{12 y}

y = \frac{1}{2} x^{2}, y = - x^{2} + 6

\int x \cdot ln (x - 1) d x

What is the integral of 10x^3+2x^2+1/2 x+3 ?
The integral of 10x^3+2x^2+1/2 x+3 is (5x^4)/2+(2x^3)/3+(x^2)/4+3x+C