What is the integral of x/2 sqrt(5x+1) ?

The integral of x/2 sqrt(5x+1) is 1/375 (15x-2)(5x+1)^{3/2}+C

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integral of x/2 sqrt(5x+1)

\int \frac{x}{2} 5 x + 1 d x

\frac{1}{375} (15 x - 2) (5 x + 1)^{\frac{3}{2}} + C

Solution steps

\int \frac{x}{2} 5 x + 1 d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x 5 x + 1 d x

Apply u-substitution

: \int \frac{u ( u - 1 )}{25} d u

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

Take the constant out:

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

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

Expand

u (u - 1) : u^{\frac{3}{2}} - u

= \frac{1}{2} \cdot \frac{1}{25} \cdot \int u^{\frac{3}{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

= \frac{1}{2} \cdot \frac{1}{25} (\int u^{\frac{3}{2}} d u - \int u d u)

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

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

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

Substitute back

u = 5 x + 1

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

Simplify

\frac{1}{2} \cdot \frac{1}{25} (\frac{2}{5} (5 x + 1)^{\frac{5}{2}} - \frac{2}{3} (5 x + 1)^{\frac{3}{2}}) : \frac{1}{375} (15 x - 2) (5 x + 1)^{\frac{3}{2}}

= \frac{1}{375} (15 x - 2) (5 x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{375} (15 x - 2) (5 x + 1)^{\frac{3}{2}} + C

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What is the integral of x/2 sqrt(5x+1) ?
The integral of x/2 sqrt(5x+1) is 1/375 (15x-2)(5x+1)^{3/2}+C