What is the integral of (3x^2)/(4sqrt(5-x^3)) ?

The integral of (3x^2)/(4sqrt(5-x^3)) is -1/2 sqrt(5-x^3)+C

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integral of (3x^2)/(4sqrt(5-x^3))

\int \frac{3 x ^{2}}{4 5 - x ^{3}} d x

- \frac{1}{2} 5 - x^{3} + C

Solution steps

\int \frac{3 x ^{2}}{4 5 - x ^{3}} d x

Take the constant out:

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

= \frac{3}{4} \cdot \int \frac{x ^{2}}{5 - x ^{3}} d x

Apply u-substitution

: \int - \frac{2}{3} d u

= \frac{3}{4} \cdot \int - \frac{2}{3} d u

Integral of a constant:

\int a d x = a x

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

Substitute back

u = 5 - x^{3}

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

Simplify

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

= - \frac{1}{2} 5 - x^{3}

Add a constant to the solution

= - \frac{1}{2} 5 - x^{3} + C

f (x) = 3 x^{2} - 3 x, at x = 3

x \to 0 lim (\frac{sinh ( x )}{x})

\frac{6 x ^{2} + 2 x + 4}{x}

y^{^{''}} + 3 y^{^{'}} - 2 y = 0

\frac{d y}{d x} = 4 e^{x - y}

What is the integral of (3x^2)/(4sqrt(5-x^3)) ?
The integral of (3x^2)/(4sqrt(5-x^3)) is -1/2 sqrt(5-x^3)+C