# Find the general indefinite integral : \int v(v^3+5)^2dv

## Question:

Find the general indefinite integral : {eq}\int v(v^3+5)^2dv {/eq}

## Integration:

We have an indefinite integral whose integrand is a polynomial. We will use the algebraic identity and then apply the standard integral formula. We will then apply the constant of integration.

## Answer and Explanation:

$$\int v(v^3+5)^2dv\\ $$

We will apply the algebraic identity to simplify the integrand:

$$\int v(v^6+25+10v^3)dv\\ \int (v^7+25v+10v^4)dv\\ $$

We will now apply the standard integral formula:

$$\frac{v^8}{8}+\frac{25v^2}{2}+2v^5+C\\ $$

Where C is the constant of integration.

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