What is the integral of 4xsqrt(2-x) ?

The integral of 4xsqrt(2-x) is -4(-2/5 (2-x)^{5/2}+4/3 (2-x)^{3/2})+C

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integral of 4xsqrt(2-x)

\int 4 x 2 - x d x

- 4 (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}}) + C

Solution steps

\int 4 x 2 - x d x

Take the constant out:

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

= 4 \cdot \int x 2 - x d x

Apply u-substitution

: \int - (- u + 2) u d u

= 4 \cdot \int - (- u + 2) u d u

Take the constant out:

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

= 4 (- \int (- u + 2) u d u)

Expand

(- u + 2) u : - u^{\frac{3}{2}} + 2 u

= 4 (- \int - u^{\frac{3}{2}} + 2 u d u)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (- (- \int u^{\frac{3}{2}} d u + \int 2 u d u))

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

= 4 (- (- \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}}))

Substitute back

u = 2 - x

= 4 (- (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}}))

Simplify

= - 4 (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}})

Add a constant to the solution

= - 4 (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}}) + C

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What is the integral of 4xsqrt(2-x) ?
The integral of 4xsqrt(2-x) is -4(-2/5 (2-x)^{5/2}+4/3 (2-x)^{3/2})+C