What is the integral from 0 to 3 of (4xf(x^2)+f(3x)) ?

The integral from 0 to 3 of (4xf(x^2)+f(3x)) is (27f)/2+81f

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integral from 0 to 3 of (4xf(x^2)+f(3x))

\int_{0}^{3} (4 x f (x^{2}) + f (3 x)) d x

\frac{27 f}{2} + 81 f

Solution steps

\int_{0}^{3} 4 x f x^{2} + f \cdot 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

= \int_{0}^{3} 3 f x d x + \int_{0}^{3} 4 f x x^{2} d x

\int_{0}^{3} 3 f x d x = \frac{27 f}{2}

\int_{0}^{3} 4 f x x^{2} d x = 81 f

= \frac{27 f}{2} + 81 f

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What is the integral from 0 to 3 of (4xf(x^2)+f(3x)) ?
The integral from 0 to 3 of (4xf(x^2)+f(3x)) is (27f)/2+81f