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

The integral of (3a-2x)^2 is 4x^2a-6xa^2+3a^3+C

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integral of (3a-2x)^2

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

4 x^{2} a - 6 x a^{2} + 3 a^{3} + C

Solution steps

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

- 2 x + 3 a = 2 x (- 1 + \frac{3 a}{2 x})

= \int (2 x (- 1 + \frac{3 a}{2 x}))^{2} d a

Simplify

(2 x (- 1 + \frac{3 a}{2 x}))^{2} : 4 x^{2} - 12 x a + 9 a^{2}

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

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^{2} d a - \int 12 x a d a + \int 9 a^{2} d a

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

\int 12 x a d a = 6 x a^{2}

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

= 4 x^{2} a - 6 x a^{2} + 3 a^{3}

Add a constant to the solution

= 4 x^{2} a - 6 x a^{2} + 3 a^{3} + C

\int (\frac{9 - x ^{2}}{x}) d x

\int \frac{t ^{3} + 5 t ^{2} + 7 t + 1}{t ^{2} + 3 t} d t

\int \frac{( 2 x ^{3} + 1 )}{x} d x

\int 18 cos (18 x) d x

\int (4 x^{3} + 1)^{3} (12 x) d x

What is the integral of (3a-2x)^2 ?
The integral of (3a-2x)^2 is 4x^2a-6xa^2+3a^3+C