What is the integral of |x-5| ?

The integral of |x-5| is ((x-5)|x-5|)/2+C

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integral of |x-5|

\int ∣ x - 5 ∣ d x

\frac{( x - 5 ) ∣ x - 5 ∣}{2} + C

Solution steps

\int ∣ x - 5 ∣ d x

Apply u-substitution

: \int ∣ u ∣ d u

= \int ∣ u ∣ d u

Use the common integral:

\int ∣ u ∣ d u = \frac{u ∣ u ∣}{2}

= \frac{u ∣ u ∣}{2}

Substitute back

u = x - 5

= \frac{( x - 5 ) ∣ x - 5 ∣}{2}

Add a constant to the solution

= \frac{( x - 5 ) ∣ x - 5 ∣}{2} + C

x \to 0 lim (\frac{1}{ln ∣ x ∣})

\int \frac{15 x e ^{2 x}}{( 1 + 2 x ) ^{2}} d x

\frac{d y}{d x}, e^{5 y^{2}} = x^{3} + 3

y^{^{'}} = (x - 2) e^{- 2 y}, y (2) = ln (2)

\frac{\partial}{\partial y} (x e^{\frac{x}{y}})