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

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

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

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

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

Solution steps

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

Expand

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

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

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 y^{2} d y + \int 10 y^{3} d y + \int y^{4} d y

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

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

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

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

Add a constant to the solution

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

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