What is the integral of 4xsqrt(5x-2) ?

The integral of 4xsqrt(5x-2) is 4/25 (2/5 (5x-2)^{5/2}+4/3 (5x-2)^{3/2})+C

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integral of 4xsqrt(5x-2)

\int 4 x 5 x - 2 d x

\frac{4}{25} (\frac{2}{5} (5 x - 2)^{\frac{5}{2}} + \frac{4}{3} (5 x - 2)^{\frac{3}{2}}) + C

Solution steps

\int 4 x 5 x - 2 d x

Take the constant out:

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

= 4 \cdot \int x 5 x - 2 d x

Apply u-substitution

: \int \frac{u ( u + 2 )}{25} d u

= 4 \cdot \int \frac{u ( u + 2 )}{25} d u

Take the constant out:

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

= 4 \cdot \frac{1}{25} \cdot \int u (u + 2) d u

Expand

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

= 4 \cdot \frac{1}{25} \cdot \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 \cdot \frac{1}{25} (\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 \cdot \frac{1}{25} (\frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}})

Substitute back

u = 5 x - 2

= 4 \cdot \frac{1}{25} (\frac{2}{5} (5 x - 2)^{\frac{5}{2}} + \frac{4}{3} (5 x - 2)^{\frac{3}{2}})

Simplify

4 \cdot \frac{1}{25} (\frac{2}{5} (5 x - 2)^{\frac{5}{2}} + \frac{4}{3} (5 x - 2)^{\frac{3}{2}}) : \frac{4}{25} (\frac{2}{5} (5 x - 2)^{\frac{5}{2}} + \frac{4}{3} (5 x - 2)^{\frac{3}{2}})

= \frac{4}{25} (\frac{2}{5} (5 x - 2)^{\frac{5}{2}} + \frac{4}{3} (5 x - 2)^{\frac{3}{2}})

Add a constant to the solution

= \frac{4}{25} (\frac{2}{5} (5 x - 2)^{\frac{5}{2}} + \frac{4}{3} (5 x - 2)^{\frac{3}{2}}) + C

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What is the integral of 4xsqrt(5x-2) ?
The integral of 4xsqrt(5x-2) is 4/25 (2/5 (5x-2)^{5/2}+4/3 (5x-2)^{3/2})+C