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

The integral of x^2(5+x)^2 is (25x^3)/3+(5x^4)/2+(x^5)/5+C

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

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

\frac{25 x ^{3}}{3} + \frac{5 x ^{4}}{2} + \frac{x ^{5}}{5} + C

Solution steps

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

Expand

x^{2} (5 + x)^{2} : 25 x^{2} + 10 x^{3} + x^{4}

= \int 25 x^{2} + 10 x^{3} + x^{4} 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 25 x^{2} d x + \int 10 x^{3} d x + \int x^{4} d x

\int 25 x^{2} d x = \frac{25 x ^{3}}{3}

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

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

= \frac{25 x ^{3}}{3} + \frac{5 x ^{4}}{2} + \frac{x ^{5}}{5}

Add a constant to the solution

= \frac{25 x ^{3}}{3} + \frac{5 x ^{4}}{2} + \frac{x ^{5}}{5} + C

\int x^{4} + \frac{3}{2} x^{2} + c d x

\int (\frac{x ^{3}}{x ^{4} + 9}) d x

\int [x (x^{2} - 16)^{2} - \frac{1}{2 x + 5}] d x

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

What is the integral of x^2(5+x)^2 ?
The integral of x^2(5+x)^2 is (25x^3)/3+(5x^4)/2+(x^5)/5+C