What is the integral of |x-2|30 ?

The integral of |x-2|30 is 15|x-2|(x-2)+C

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integral of |x-2|30

integral

∣ x - 2 ∣ 30

15 ∣ x - 2 ∣ (x - 2) + C

Solution steps

\int ∣ x - 2 ∣ \cdot 30 d x

Take the constant out:

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

= 30 \cdot \int ∣ x - 2 ∣ d x

Apply u-substitution

: \int ∣ u ∣ d u

= 30 \cdot \int ∣ u ∣ d u

Use the common integral:

\int ∣ u ∣ d u = \frac{u ∣ u ∣}{2}

= 30 \cdot \frac{u ∣ u ∣}{2}

Substitute back

u = x - 2

= 30 \cdot \frac{( x - 2 ) ∣ x - 2 ∣}{2}

Simplify

30 \cdot \frac{( x - 2 ) ∣ x - 2 ∣}{2} : 15 ∣ x - 2 ∣ (x - 2)

= 15 ∣ x - 2 ∣ (x - 2)

Add a constant to the solution

= 15 ∣ x - 2 ∣ (x - 2) + C

\frac{1}{x ^{6}}

2 e^{u}

e^{3 x^{2}}

x^{4} \cdot (e^{- x})^{2}

\frac{y}{( 1 + y ^{2} )}