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

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

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integral of (x^2+x)^2

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

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

Solution steps

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

Expand

(x^{2} + x)^{2} : x^{4} + 2 x^{3} + x^{2}

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

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

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

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

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

Add a constant to the solution

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

n = 0 \sum \infty \frac{1}{2 ( 2 n + 1 )}

\int (x^{7} - 2 x^{6})^{4} (7 x^{6} - 12 x^{5}) d x

x \to 0 lim (\frac{50 x ^{2}}{sin ( x ) + 50 x ^{2}})

\int_{0}^{1} e^{2 x} d x

f (x) = x^{3} - 2 x - 2, at x = - 1

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