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

The integral of 3x(5x+2)^3 is 3(25x^5+(75x^4)/2+20x^3+4x^2)+C

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integral of 3x(5x+2)^3

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

3 (25 x^{5} + \frac{75 x ^{4}}{2} + 20 x^{3} + 4 x^{2}) + C

Solution steps

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

Take the constant out:

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

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

Expand

x (5 x + 2)^{3} : 125 x^{4} + 150 x^{3} + 60 x^{2} + 8 x

= 3 \cdot \int 125 x^{4} + 150 x^{3} + 60 x^{2} + 8 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

= 3 (\int 125 x^{4} d x + \int 150 x^{3} d x + \int 60 x^{2} d x + \int 8 x d x)

\int 125 x^{4} d x = 25 x^{5}

\int 150 x^{3} d x = \frac{75 x ^{4}}{2}

\int 60 x^{2} d x = 20 x^{3}

\int 8 x d x = 4 x^{2}

= 3 (25 x^{5} + \frac{75 x ^{4}}{2} + 20 x^{3} + 4 x^{2})

Add a constant to the solution

= 3 (25 x^{5} + \frac{75 x ^{4}}{2} + 20 x^{3} + 4 x^{2}) + C

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What is the integral of 3x(5x+2)^3 ?
The integral of 3x(5x+2)^3 is 3(25x^5+(75x^4)/2+20x^3+4x^2)+C