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

The integral of (x^3+x)^2 is (x^7)/7+(2x^5)/5+(x^3)/3+C

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integral of (x^3+x)^2

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

\frac{x ^{7}}{7} + \frac{2 x ^{5}}{5} + \frac{x ^{3}}{3} + C

Solution steps

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

Expand

(x^{3} + x)^{2} : x^{6} + 2 x^{4} + x^{2}

= \int x^{6} + 2 x^{4} + 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^{6} d x + \int 2 x^{4} d x + \int x^{2} d x

\int x^{6} d x = \frac{x ^{7}}{7}

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

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

= \frac{x ^{7}}{7} + \frac{2 x ^{5}}{5} + \frac{x ^{3}}{3}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{2 x ^{5}}{5} + \frac{x ^{3}}{3} + C

d x - (x + y) d y = 0

\int cos (4 y) d y

\int x^{2} x^{2} - 25 d x

y^{^{'}} + \frac{1}{t} y = 0

n = 1 \sum \infty (\frac{x}{n})^{n}

What is the integral of (x^3+x)^2 ?
The integral of (x^3+x)^2 is (x^7)/7+(2x^5)/5+(x^3)/3+C