What is the integral of 3(sqrt(2x)-x+4) ?

The integral of 3(sqrt(2x)-x+4) is 2sqrt(2)x^{3/2}-(3x^2)/2+12x+C

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integral of 3(sqrt(2x)-x+4)

\int 3 (2 x - x + 4) d x

22 x^{\frac{3}{2}} - \frac{3 x ^{2}}{2} + 12 x + C

Solution steps

\int 3 (2 x - x + 4) d x

Take the constant out:

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

= 3 \cdot \int 2 x - x + 4 d x

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 3 \cdot \int 2 x - x + 4 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 2 x d x - \int x d x + \int 4 d x)

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

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

\int 4 d x = 4 x

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

Simplify

3 (2 \frac{2}{3} x^{\frac{3}{2}} - \frac{x ^{2}}{2} + 4 x) : 22 x^{\frac{3}{2}} - \frac{3 x ^{2}}{2} + 12 x

= 22 x^{\frac{3}{2}} - \frac{3 x ^{2}}{2} + 12 x

Add a constant to the solution

= 22 x^{\frac{3}{2}} - \frac{3 x ^{2}}{2} + 12 x + C

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What is the integral of 3(sqrt(2x)-x+4) ?
The integral of 3(sqrt(2x)-x+4) is 2sqrt(2)x^{3/2}-(3x^2)/2+12x+C