What is the integral of x+(x^2)/2+(5x^3)/3 ?

The integral of x+(x^2)/2+(5x^3)/3 is (x^2)/2+(x^3)/6+(5x^4)/(12)+C

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integral of x+(x^2)/2+(5x^3)/3

\int x + \frac{x ^{2}}{2} + \frac{5 x ^{3}}{3} d x

\frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \frac{5 x ^{4}}{12} + C

Solution steps

\int x + \frac{x ^{2}}{2} + \frac{5 x ^{3}}{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 x d x + \int \frac{x ^{2}}{2} d x + \int \frac{5 x ^{3}}{3} d x

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

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

\int \frac{5 x ^{3}}{3} d x = \frac{5 x ^{4}}{12}

= \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \frac{5 x ^{4}}{12}

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \frac{5 x ^{4}}{12} + C

What is the integral of x+(x^2)/2+(5x^3)/3 ?
The integral of x+(x^2)/2+(5x^3)/3 is (x^2)/2+(x^3)/6+(5x^4)/(12)+C