What is the integral from 3 to 6 of (2ax+3)3x ?

The integral from 3 to 6 of (2ax+3)3x is 3(126a+81/2)

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integral from 3 to 6 of (2ax+3)3x

\int_{3}^{6} (2 a x + 3) 3 x d x

3 (126 a + \frac{81}{2})

Solution steps

\int_{3}^{6} (2 a x + 3) \cdot 3 x d x

Take the constant out:

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

= 3 \cdot \int_{3}^{6} (2 a x + 3) x d x

Expand

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

= 3 \cdot \int_{3}^{6} 2 a x^{2} + 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

= 3 (\int_{3}^{6} 2 a x^{2} d x + \int_{3}^{6} 3 x d x)

\int_{3}^{6} 2 a x^{2} d x = 126 a

\int_{3}^{6} 3 x d x = \frac{81}{2}

= 3 (126 a + \frac{81}{2})

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What is the integral from 3 to 6 of (2ax+3)3x ?
The integral from 3 to 6 of (2ax+3)3x is 3(126a+81/2)