What is the integral of xsqrt(15x-5) ?

The integral of xsqrt(15x-5) is 2/(27sqrt(5))(9x+2)(3x-1)^{3/2}+C

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integral of xsqrt(15x-5)

\int x 15 x - 5 d x

\frac{2}{27 5} (9 x + 2) (3 x - 1)^{\frac{3}{2}} + C

Solution steps

\int x 15 x - 5 d x

Apply u-substitution

: \int \frac{u ( u + 5 )}{225} d u

= \int \frac{u ( u + 5 )}{225} d u

Take the constant out:

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

= \frac{1}{225} \cdot \int u (u + 5) d u

Expand

u (u + 5) : u^{\frac{3}{2}} + 5 u

= \frac{1}{225} \cdot \int u^{\frac{3}{2}} + 5 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}{225} (\int u^{\frac{3}{2}} d u + \int 5 u d u)

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

\int 5 u d u = \frac{10}{3} u^{\frac{3}{2}}

= \frac{1}{225} (\frac{2}{5} u^{\frac{5}{2}} + \frac{10}{3} u^{\frac{3}{2}})

Substitute back

u = 15 x - 5

= \frac{1}{225} (\frac{2}{5} (15 x - 5)^{\frac{5}{2}} + \frac{10}{3} (15 x - 5)^{\frac{3}{2}})

Simplify

\frac{1}{225} (\frac{2}{5} (15 x - 5)^{\frac{5}{2}} + \frac{10}{3} (15 x - 5)^{\frac{3}{2}}) : \frac{2}{27 5} (9 x + 2) (3 x - 1)^{\frac{3}{2}}

= \frac{2}{27 5} (9 x + 2) (3 x - 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{27 5} (9 x + 2) (3 x - 1)^{\frac{3}{2}} + C

\frac{1}{x} d x - \frac{1}{y} d y = 0

e^{x} - x^{5}

\frac{y}{2 ( x + y )} = \frac{d y}{d x}

f (x) = (3 x^{5} + x 6^{x}) 2^{32}

(1, - 9), (- 3, - 9)

What is the integral of xsqrt(15x-5) ?
The integral of xsqrt(15x-5) is 2/(27sqrt(5))(9x+2)(3x-1)^{3/2}+C