What is the antiderivative of (5 + 6x)^2?

Expert Answers

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You may use substitution as alternative method, hence you should come up with the notation `5 + 6x = y` .

Differentiating `5 + 6x = y`  yields:

`6dx = dy =gt dx = dy/6`

You need to change the variable under integral such that:

`int (5+6x)^2 dx = int (t^2)(dy/6)`

`int (t^2)(dy/6) = (1/6)*(t^3)/3 + c`

`int (t^2)(dy/6) = t^3/18 + c`

`int (5+6x)^2 dx = ((5+6x)^3)/18 + c`

Hence, evaluating the integral yields `int (5+6x)^2 dx = ((5+6x)^3)/18 + c.`

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The anti derivative of (5 + 6x)^2 has to be determined.

`int (5 + 6x)^2 dx`

=> `int 25 + 36x^2 + 60x dx`

=> `25x + 36x^3/3 + 60x^2/2`

=> `25x + 12x^3 + 30x^2`

The anti derivative of `(5 + 6x)^2` is `12x^3 + 30x^2 + 25x + C`

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