Solve. ∫(4ax3+3bx2+2cx)dx

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ottcomn · Accepted Answer

There are three properties of integrals that we'll exploit to evaluate this. Those are
1.The integral of a sum is the sum of the integrals of the terms.
$\int [f (x) + g (x)] d x = \int f (x) d x + \int g (x) d x$
2.The integral of a constant times a function is equal to the constant times the integral of the function.
$\int k f (x) d x = k \int f (x) d x$
3.The integral of a power n of x is given by
$\int x^{n} d x = (x^{(} n + 1)) / (n + 1) + C$
Now let's use those. Firstly, we use property 1 to split up that integral.
$\int (4 a x^{3} + 3 b x^{2} + 2 c x) d x = \int 4 a x^{3} d x + \int 3 b x^{2} d x + \int 2 c x d x$
Next we'll use property 2 to pull out any constants.
$4 a \int x^{3} d x + 3 b \int x^{2} d x + 2 c \int x d x$
And finally we'll use property 3 to evaluate those integrals.
$= 4 a (x^{4}) / 4 + 3 b (x^{3}) / 3 + 2 c (x^{2}) / 2 + C$
Now we just simplify and we're done!
$a x^{4} + b x^{3} + c x^{2} + C$

Solve. int (4ax^3+3bx^2+2cx)dx

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