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

The integral of (2sqrt(x)-3x)^2 is 2x^2-24/5 x^{5/2}+3x^3+C

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integral of (2sqrt(x)-3x)^2

\int (2 x - 3 x)^{2} d x

2 x^{2} - \frac{24}{5} x^{\frac{5}{2}} + 3 x^{3} + C

Solution steps

\int (2 x - 3 x)^{2} d x

Expand

(2 x - 3 x)^{2} : 4 x - 12 x^{\frac{3}{2}} + 9 x^{2}

= \int 4 x - 12 x^{\frac{3}{2}} + 9 x^{2} 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

= \int 4 x d x - \int 12 x^{\frac{3}{2}} d x + \int 9 x^{2} d x

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

\int 12 x^{\frac{3}{2}} d x = \frac{24}{5} x^{\frac{5}{2}}

\int 9 x^{2} d x = 3 x^{3}

= 2 x^{2} - \frac{24}{5} x^{\frac{5}{2}} + 3 x^{3}

Add a constant to the solution

= 2 x^{2} - \frac{24}{5} x^{\frac{5}{2}} + 3 x^{3} + C

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

\int \frac{2}{3} t^{\frac{3}{2}} d t

\int \frac{2 t ^{2} + t ^{2} t - 1}{t ^{2}} d t

\int x^{2} (5 x^{3} - 3)^{4} d x

What is the integral of (2sqrt(x)-3x)^2 ?
The integral of (2sqrt(x)-3x)^2 is 2x^2-24/5 x^{5/2}+3x^3+C