What is integral of xf(x) ?

The answer to integral of xf(x) is (fx^3)/3+C

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integral of xf(x)

\int x f (x) d x

\frac{f x ^{3}}{3} + C

Solution steps

\int x f x d x

Take the constant out:

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

= f \cdot \int xx d x

Simplify

xx : x^{2}

= f \cdot \int x^{2} d x

Apply the Power Rule

: \frac{x ^{3}}{3}

= f \frac{x ^{3}}{3}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{f x ^{3}}{3}

Add a constant to the solution

= \frac{f x ^{3}}{3} + C

f (x) = 17 e^{x}

f (x) = - x^{2} + 4 x, at x = 1

\frac{x d y}{d x} + 5 y = x^{3} - x

\int_{0}^{1} \frac{e ^{x} + 1}{e ^{x} + x} d x

y^{^{'}} = y sin (x), y (0) = 1