Give antiderivative of y=x^2/(x^3+9)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the anti-derivative of the given function, hence, you need to find the indefinite integral of the function, such that:

`int x^2/(x^3+9) dx `

You should come up with the following substitution, such that:

`x^3 + 9 = t => 3x^2dx = dt => x^2dx = (dt)/3`

Replacing the variable yields:

`int x^2/(x^3+9) dx = int ((dt)/3)/t`

`int ((dt)/3)/t = (1/3)ln|t| + c`

Replacing back `x^3 + 9` for t yields:

`int x^2/(x^3+9) dx = (1/3)ln|x^3+9| + c`

Using the power property of logarithms yields:

`int x^2/(x^3+9) dx = ln|x^3+9|^(1/3) + c`

`int x^2/(x^3+9) dx = ln root(3)(|x^3+9|) + c`

Hence, evaluating the anti-derivative of the given function yields `int x^2/(x^3+9) dx = ln root(3)(|x^3+9|) + c` .

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