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

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

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

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

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

Solution steps

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

\int 1 d x = x

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

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

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

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

Add a constant to the solution

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

\int x e^{- \frac{x ^{2}}{4}} d x

\int 3 x - \frac{1}{3 x} d x

\int e^{(2 x^{3} - 3 x^{2} + 5)} (x^{2} - x) d x

\int (2 x^{4} - 5 x^{2} + 1) d x

\int \frac{1}{5 + 2 x - 2 x ^{2}} d x

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