What is the integral of x^2(5x^3-1)^{3/2} ?

The integral of x^2(5x^3-1)^{3/2} is 1/75 (10x^3(5x^3-1)^{3/2}-2(5x^3-1)^{3/2})+C

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integral of x^2(5x^3-1)^{3/2}

\int x^{2} (5 x^{3} - 1)^{\frac{3}{2}} d x

\frac{1}{75} (10 x^{3} (5 x^{3} - 1)^{\frac{3}{2}} - 2 (5 x^{3} - 1)^{\frac{3}{2}}) + C

Solution steps

\int x^{2} (5 x^{3} - 1)^{\frac{3}{2}} d x

Apply u-substitution

: \int \frac{u ( 5 u ^{\frac{3}{2}} - 1 ) ^{\frac{3}{2}}}{2 x} d u

= \int \frac{( 5 u - 1 ) ^{\frac{3}{2}}}{3} d u

Take the constant out:

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

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

Apply Integration By Parts

: u (5 u - 1)^{\frac{3}{2}} - \int \frac{15}{2} u (5 u - 1)^{\frac{1}{2}} d u

= \frac{1}{3} (u (5 u - 1)^{\frac{3}{2}} - \int \frac{15}{2} u (5 u - 1)^{\frac{1}{2}} d u)

\int \frac{15}{2} u (5 u - 1)^{\frac{1}{2}} d u = \frac{1}{25} (15 u + 2) (5 u - 1)^{\frac{3}{2}}

= \frac{1}{3} (u (5 u - 1)^{\frac{3}{2}} - \frac{1}{25} (15 u + 2) (5 u - 1)^{\frac{3}{2}})

Substitute back

u = x^{3}

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

Simplify

\frac{1}{3} (x^{3} (5 x^{3} - 1)^{\frac{3}{2}} - \frac{1}{25} (15 x^{3} + 2) (5 x^{3} - 1)^{\frac{3}{2}}) : \frac{1}{75} (10 x^{3} (5 x^{3} - 1)^{\frac{3}{2}} - 2 (5 x^{3} - 1)^{\frac{3}{2}})

= \frac{1}{75} (10 x^{3} (5 x^{3} - 1)^{\frac{3}{2}} - 2 (5 x^{3} - 1)^{\frac{3}{2}})

Add a constant to the solution

= \frac{1}{75} (10 x^{3} (5 x^{3} - 1)^{\frac{3}{2}} - 2 (5 x^{3} - 1)^{\frac{3}{2}}) + C

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

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (0) = - 1, y^{^{'}} (0) = 0

x \to - 4 lim (- 2 x + 3)

\frac{\partial}{\partial x} (\frac{4 x - 2 y}{2 x ^{2} + 3 y})

\frac{d}{d x} (\frac{5}{x ^{2} + 25})

What is the integral of x^2(5x^3-1)^{3/2} ?
The integral of x^2(5x^3-1)^{3/2} is 1/75 (10x^3(5x^3-1)^{3/2}-2(5x^3-1)^{3/2})+C