What is the integral of (x^2+3)5x ?

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

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integral of (x^2+3)5x

\int (x^{2} + 3) 5 x d x

5 (\frac{x ^{4}}{4} + \frac{3 x ^{2}}{2}) + C

Solution steps

\int (x^{2} + 3) \cdot 5 x d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int (x^{2} + 3) x d x

Expand

(x^{2} + 3) x : x^{3} + 3 x

= 5 \cdot \int x^{3} + 3 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 (\int x^{3} d x + \int 3 x d x)

\int x^{3} d x = \frac{x ^{4}}{4}

\int 3 x d x = \frac{3 x ^{2}}{2}

= 5 (\frac{x ^{4}}{4} + \frac{3 x ^{2}}{2})

Add a constant to the solution

= 5 (\frac{x ^{4}}{4} + \frac{3 x ^{2}}{2}) + C

f (x) = (3 x^{3} + 2) (x^{4} - 2 x)

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