What is the integral of sqrt(2a-2x) ?

The integral of sqrt(2a-2x) is -sqrt(2) 2/3 (a-x)^{3/2}+C

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integral of sqrt(2a-2x)

\int 2 a - 2 x d x

- 2 \frac{2}{3} (a - x)^{\frac{3}{2}} + C

Solution steps

\int 2 a - 2 x d x

2 a - 2 x = 2 a - x

= \int 2 a - x d x

Take the constant out:

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

= 2 \cdot \int a - x d x

Apply u-substitution

: \int - u d u

= 2 \cdot \int - u d u

Take the constant out:

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

= 2 (- \int u d u)

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= 2 (- \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = a - x

= 2 (- \frac{2}{3} (a - x)^{\frac{3}{2}})

Simplify

= - 2 \frac{2}{3} (a - x)^{\frac{3}{2}}

Add a constant to the solution

= - 2 \frac{2}{3} (a - x)^{\frac{3}{2}} + C

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What is the integral of sqrt(2a-2x) ?
The integral of sqrt(2a-2x) is -sqrt(2) 2/3 (a-x)^{3/2}+C